with Azumaya algebras over rings in which 2 is invertible, whereas [?] focuses on

central simple algebras, including the characteristic 2 case).

The description of groups of similitudes of quadratic spaces of dimension 3, 4,

5 and 6 dates back to Van der Waerden [?] and Dieudonn´ [?, §3]. The case of

e

quadratic spaces over rings was treated by Knus [?, §3] and by Knus-Parimala-

Sridharan [?, §6] (see also Knus [?, Ch. 5]). Cli¬ord groups of algebras of degree 4

or 6 with orthogonal involution are determined in Merkurjev-Tignol [?, 1.4.2, 1.4.3].

§??. Albert forms are introduced in Albert [?]. Theorem 3 of that paper yields

the criterion for the biquaternion algebra A = (a, b)F — (c, d)F to be a division alge-

bra in terms of the associated quadratic form q = a, b, ’ab, ’c, ’d, cd . (See (??).)

The de¬nition of the form q thus depends on a particular decomposition of the bi-

quaternion algebra A. The fact that quadratic forms associated to di¬erent decom-

positions are similar was ¬rst proved by Jacobson [?, Theorem 3.12] using Jordan

structures, and later by Knus [?, Proposition 1.14] and Mammone-Shapiro [?] us-

ing the algebraic theory of quadratic forms. (See also Knus-Parimala-Sridharan [?,

Theorem 4.2].) Other proofs of Albert™s Theorem (??) were given by P¬ster [?,

p. 124] and Tamagawa [?] (see also Seligman [?, App. C]). A notion of Albert form

in characteristic 2 is discussed in Mammone-Shapiro [?]. Note that the original

version of Jacobson™s paper [?] does not cover the characteristic 2 case adequately;

see the reprinted version in Jacobson™s Collected Mathematical Papers [?], where

the characteristic is assumed to be di¬erent from 2.

From the de¬nition of the quadratic form q = a, b, ’ab, ’c, ’d, cd associated

to A = (a, b)F — (c, d)F , it is clear that q is isotropic if and only if the quaternion

algebras (a, b)F and (c, d)F have a common maximal sub¬eld. Thus Corollary (??)

easily follows from Albert™s Theorem (??). The proof given by Albert in [?] is more

direct and also holds in characteristic 2. Another proof (also valid in characteris-

tic 2) was given by Sah [?]. If the characteristic is 2, the result can be made more

precise: if a tensor product of two quaternion division algebras is not a division

algebra, then the two quaternion algebras have a common maximal sub¬eld which

is a separable extension of the center. This was ¬rst shown by Draxl [?]. The proofs

given by Knus [?] and by Tits [?] work in all characteristics, and yield the more

precise result in characteristic 2.

The fact that G(q) = F —2 · NrdA (A— ) for an Albert form q of a biquaternion

algebra A is already implicit in Van der Waerden [?, pp. 21“22] and Dieudonn´ [?, e

Nos 28, 30, 34]. Knus-Lam-Shapiro-Tignol [?] gives other characterizations of that

group. In particular, it is shown that this group is also the set of discriminants of

orthogonal involutions on A; see Parimala-Sridharan-Suresh [?] for another proof

of that result.

If σ is a symplectic involution on a biquaternion algebra A, the (Albert) quad-

ratic form Nrpσ de¬ned on the space Symd(A, σ) is the generic norm of Symd(A, σ),

viewed as a Jordan algebra, see Jacobson [?]. This is the point of view from which

276 IV. ALGEBRAS OF DEGREE FOUR

results on Albert forms are derived in Jacobson [?, Ch. 6, §4]. The invariant jσ („ )

of symplectic involutions is de¬ned in a slightly di¬erent fashion in Knus-Lam-

Shapiro-Tignol [?, §3]. See (??) for the relation between jσ („ ) and Rost™s higher

cohomological invariants.

If σ is an orthogonal involution on a biquaternion algebra A, the linear endomor-

phism pσ of Skew(A, σ) was ¬rst de¬ned by Knus-Parimala-Sridharan [?], [?] (see

also Knus [?, Ch. 5]), who called it the pfa¬an adjoint because of its relation with

the pfa¬an. Pfa¬an adjoints for algebras of degree greater than 4 are considered

in Knus-Parimala-Sridharan [?]. If deg A = 2m, the pfa¬an adjoint is a polynomial

map of degree m ’ 1 from Skew(A, σ) to itself. Knus-Parimala-Sridharan actually

treat orthogonal and symplectic involutions simultaneously (and in the context of

Azumaya algebras): if σ is a symplectic involution on a biquaternion algebra A,

the pfa¬an adjoint is the endomorphism of Sym(A, σ) de¬ned in §??.

Further results on Albert forms can be found in Lam-Leep-Tignol [?], where

maximal sub¬elds of a biquaternion algebra are investigated; in particular, neces-

sary and su¬cient conditions for the cyclicity of a biquaternion algebra are given

in that paper.

§??. Suppose char F = 2. As observed in the proof of (??), the group “ =

{ (», a) ∈ F — —A— | »2 = NrdA (a) }, for A a biquaternion F -algebra, can be viewed

as the special Cli¬ord group of the quadratic space Sym(A, σ), Nrpσ for any

symplectic involution σ. The map ±σ : “ ’ I 4 F/I 5 F can actually be de¬ned on the

full Cli¬ord group “ Sym(A, σ), Nrpσ by mapping the generators v ∈ Sym(A, σ)—

to ¦v + I 5 F . Showing that this map is well-de¬ned is the main di¬culty of this

approach.

The homomorphism ± : SK1 (A) ’ I 4 F/I 5 F (for A a biquaternion algebra) was

originally de¬ned by Rost in terms of Galois cohomology, as a map ± : SK1 (A) ’

H 4 (F, µ2 ). Rost also proved exactness of the following sequence:

± h

0 ’ SK1 (A) ’ H 4 (F, µ2 ) ’ H 4 (F (q), µ2 )

’ ’

where h is induced by scalar extension to the function ¬eld of an Albert form

q, see Merkurjev [?, Theorem 4] and Kahn-Rost-Sujatha [?]. The point of view

of quadratic forms developed in §?? is equivalent, in view of the isomorphism

∼

e4 : I 4 F/I 5 F ’ H 4 (F, µ2 ) proved by Rost (unpublished) and more recently by

’

Voevodsky, which leads to a commutative diagram

± i

0 ’ ’ ’ SK1 (A) ’ ’ ’ I 4 F/I 5 F ’ ’ ’ I 4 F (q)/I 5 F (q)

’’ ’’ ’’

¦ ¦

¦ ¦

e e

4 4

± h

0 ’ ’ ’ SK1 (A) ’ ’ ’ H 4 (F, µ2 ) ’ ’ ’ H 4 F (q), µ2 .

’’ ’’ ’’

The fact that the sequences above are zero-sequences readily follows from functo-

riality of ± and ± , since SK1 (A) = 0 if A is not a division algebra or, equivalently

by (??), if q is isotropic. In order to derive exactness of the sequence above from

exactness of the sequence below, only “elementary” information on the map e 4 is

needed: it is su¬cient to use the fact that on P¬ster forms e4 is well-de¬ned (see

Elman-Lam [?, 3.2]) and injective (see Arason-Elman-Jacob [?, Theorem 1]). In

fact, (??) shows that the image of ± consists of 4-fold P¬ster forms (modulo I 5 F )

and, on the other hand, the following proposition also shows that every element in

ker i is represented by a single 4-fold P¬ster form:

NOTES 277

(17.30) Proposition. If q is an Albert form which represents 1, then

ker i = { π + I 5 F | π is a 4-fold P¬ster form containing q }.

Proof : By Fitzgerald [?, Corollary 2.3], the kernel of the scalar extension map

W F ’ W F (q) is an ideal generated by 4-fold and 5-fold P¬ster forms. Therefore,

every element in ker i is represented by a sum of 4-fold P¬ster forms which become

hyperbolic over F (q). By the Cassels-P¬ster subform theorem (see Scharlau [?,

Theorem 4.5.4]) the P¬ster forms which satisfy this condition contain a subform

isometric to q. If π1 , π2 are two such 4-fold P¬ster forms, then the Witt index of

π1 ⊥ ’π2 is at least dim q = 6. By Elman-Lam [?, Theorem 4.5] it follows that

and elements a1 , a2 ∈ F — such that

there exists a 3-fold P¬ster form

π1 = · a1 and π2 = · a2 .

Therefore,

mod I 5 F.

π1 + π 2 ≡ · a 1 a2

By induction on the number of terms, it follows that every sum of 4-fold P¬ster

forms representing an element of ker i is equivalent modulo I 5 F to a single 4-fold

P¬ster form.

The image of the map ±σ can be described in a similar way. Since ±σ = 0 if σ

is hyperbolic or, equivalently by (??), if the 5-dimensional form sσ is isotropic, it

follows by functoriality of ±σ that im ±σ lies in the kernel of the scalar extension

map to F (sσ ). One can use the arguments in the proof of Merkurjev [?, Theorem 4]

to show that this inclusion is an equality, so that the following sequence is exact:

±

“σ ’’ I 4 F/I 5 F ’’ I 4 F (sσ )/I 5 F (sσ ).

σ

’

Corollary (??) shows that every element in [A— , A— ] is a product of two com-

mutators. If A is split, every element in [A— , A— ] can actually be written as a single

commutator, as was shown by Thompson [?]. On the other hand, Kursov [?] has

found an example of a biquaternion algebra A such that the group [A— , A— ] does

not consist of commutators, hence our lower bound for the number of factors is

sharp, in general.

The ¬rst example of a biquaternion algebra A such that SK1 (A) = 0 is due

to Platonov [?]. For a slightly di¬erent relation between the reduced Whitehead

group of division algebras and Galois cohomology of degree 4, see Suslin [?].

Along with the group Σσ (A), the group generated by skew-symmetric units in a

central simple algebra with involution (A, σ) is also discussed in Yanchevski˜ [?]. If ±

—

σ is orthogonal, it is not known whether this subgroup is normal in A . Triviality of

the group USK1 (B, „ ) can be shown not only for division algebras of prime degree,

but also for division algebras whose degree is square-free; indeed, the exponent of

USK1 (B, „ ) divides deg B/p1 . . . pr , where p1 , . . . , pr are the prime factors of deg B:

see Yanchevski˜ [?]. Examples where the group K1 Spin(A) is not trivial are given

±

in Monastyrny˜ ±-Yanchevski˜ [?]. See also Yanchevski˜ [?] for the relation between

± ±

K1 Spin and decomposability of involutions.

278 IV. ALGEBRAS OF DEGREE FOUR

CHAPTER V

Algebras of Degree Three

The main topic of this chapter is central simple algebras of degree 3 with

involutions of the second kind and their ´tale (commutative) subalgebras. To every

e

involution of the second kind on a central simple algebra of degree 3, we attach

a 3-fold P¬ster form, and we show that this quadratic form classi¬es involutions

up to conjugation. Involutions whose associated P¬ster form is hyperbolic form

a distinguished conjugacy class; we show that such an involution is present on

every central simple algebra of degree 3 with involution of the second kind, and we

characterize distinguished involutions in terms of ´tale subalgebras of symmetric

e

elements.

We start with Galois descent, followed by a general discussion of ´tale and

e

Galois algebras. These are tools which will also be used in subsequent chapters.

´

§18. Etale and Galois Algebras

Throughout this section, let F be an arbitrary base ¬eld, let Fsep be a separable

closure of F and let “ be the absolute Galois group of F :

“ = Gal(Fsep /F ).

The central theme of this section is a correspondence between ´tale F -algebras

e

and “-sets, which is set up in the ¬rst subsection. This correspondence is then

restricted to Galois algebras and torsors. The ¬nal subsection demonstrates the

special features of ´tale algebras of dimension 3.

e

The key tool for the correspondence between ´tale F -algebras and “-sets is the

e

following Galois descent principle. Let V0 be a vector space over F . The left action

of “ on V = V0 — Fsep de¬ned by γ — (u — x) = u — γ(x) for u ∈ V0 and x ∈ Fsep is

semilinear with respect to “, i.e.,

γ — (vx) = (γ — v)γ(x)

for v ∈ V and x ∈ Fsep ; the action is also continuous since for every vector v ∈ V

there is a ¬nite extension M of F in Fsep such that Gal(Fsep /M ) acts trivially on v.

The space V0 can be recovered as the set of ¬xed elements of V under “. More

generally:

(18.1) Lemma (Galois descent). Let V be a vector space over Fsep . If “ acts

continuously on V by semilinear automorphisms, then

V “ = { v ∈ V | γ — v = v for all γ ∈ “ }

is an F -vector space and the map V “ — Fsep ’ V , v — x ’ vx, is an isomorphism

of Fsep -vector spaces.

279

280 V. ALGEBRAS OF DEGREE THREE

Proof : It is clear that V “ is an F -vector space. To prove surjectivity of the canon-

ical map V “ — Fsep ’ V , consider an arbitrary vector v ∈ V . Since the action of

“ on V is continuous, we may ¬nd a ¬nite Galois extension M of F in Fsep such

that Gal(Fsep /M ) acts trivially on v. Let (mi )1¤i¤n be a basis of M over F and

let (γi )1¤i¤n be a set of representatives of the left cosets of Gal(Fsep /M ) in “, so

that the orbit of v in V consists of γ1 — v, . . . , γn — v, with γ1 — v = v, say. Let

n

vj = (γi — v)γi (mj ) for j = 1, . . . , n.

i=1

Since for every γ ∈ “ and i ∈ {1, . . . , n} there exists ∈ {1, . . . , n} and γ ∈

Gal(Fsep /M ) such that γγi = γ γ , the action of γ on the right-hand side of the

expression above merely permutes the terms of the sum, hence vj ∈ V “ for j =

1, . . . , n. On the other hand, the (n — n) matrix γi (mj ) 1¤i,j¤n with entries in

M is invertible, since γ1 , . . . , γn are linearly independent over M in EndF (M )

(Dedekind™s lemma). If (mij )1¤i,j¤n is the inverse matrix, we have

n

v = γ1 — v = vi mi1 ,

i=1

hence v lies in the image of the canonical map V “ — Fsep ’ V .

To prove injectivity of the canonical map, it su¬ces to show that F -linearly

independent vectors in V “ are mapped to Fsep -linearly independent vectors in V .

Suppose the contrary; let v1 , . . . , vr ∈ V “ be F -linearly independent vectors for

which there exist nonzero elements m1 , . . . , mr ∈ Fsep such that r vi mi =

i=1

0. We may assume r is minimal, r > 1, and m1 = 1. The mi are not all in

F , hence we may assume m2 ∈ F . Choose γ ∈ “ satisfying γ(m2 ) = m2 . By

applying γ to both sides of the linear dependence relation and subtracting, we

r

obtain i=2 vi γ(mi ) ’ mi = 0, a nontrivial relation with fewer terms. This

contradiction proves that the canonical map V “ — Fsep ’ V is injective.

(18.2) Remark. Assume that V in Lemma (??) admits an Fsep -bilinear multipli-

cation m : V —V ’ V and that “ acts on V by (semilinear) algebra automorphisms;

then the restriction of m to V “ is a multiplication on V “ , hence V “ is an F -algebra.

Similarly, if V = V1 ⊃ V2 ⊃ · · · ⊃ Vr is a ¬nite ¬‚ag in V , i.e., Vi is a subspace

of Vj for i > j, and the action of “ on V preserves V2 , . . . , Vr , then the ¬‚ag in V

descends to a ¬‚ag V “ = V1“ ⊃ V2“ ⊃ · · · ⊃ Vr“ in V “ .

´

18.A. Etale algebras. Let Alg F be the category of unital commutative asso-

ciative F -algebras with F -algebra homomorphisms as morphisms. For every ¬nite

dimensional commutative F -algebra L, let X(L) be the set of F -algebra homomor-

phisms from L to Fsep :

X(L) = HomAlg F (L, Fsep ).

For any ¬eld extension K/F , let LK be the K-algebra L —F K. If K ‚ Fsep ,

then Fsep also is a separable closure of K, and every F -algebra homomorphism

L ’ Fsep extends in a unique way to a K-algebra homomorphism LK ’ Fsep ; we

may therefore identify:

X(LK ) = X(L).

The following proposition characterizes ´tale F -algebras:

e

´

§18. ETALE AND GALOIS ALGEBRAS 281

(18.3) Proposition. For a ¬nite dimensional commutative F -algebra L, the fol-

lowing conditions are equivalent:

(1) for every ¬eld extension K/F , the K-algebra LK is reduced, i.e., LK does not

contain any nonzero nilpotent elements;

(2) L K1 — · · · — Kr for some ¬nite separable ¬eld extensions K1 , . . . , Kr of F ;

(3) LFsep Fsep — · · · — Fsep ;

(4) the bilinear form T : L — L ’ F induced by the trace:

T (x, y) = TL/F (xy) for x, y ∈ L

is nonsingular;

(5) card X(L) = dimF L;

(6) card X(L) ≥ dimF L.

If the ¬eld F is in¬nite, the conditions above are also equivalent to:

(7) L F [X]/(f ) for some polynomial f ∈ F [X] which has no multiple root in an

algebraic closure of F .

References: The equivalences (??) ” (??) ” (??) are proven in Bourbaki [?, Th´o- e

r`me 4, p. V.34] and (??) ” (??) ” (??) in [?, Corollaire, p. V.29]. The equivalence

e

of (??) with the other conditions is shown in [?, Proposition 1, p. V.47]. (See also

Waterhouse [?, §6.2] for the equivalence of (??), (??), (??), and (??)). Finally,

to see that (??) characterizes ´tale algebras over an in¬nite ¬eld, see Bourbaki [?,

e

Proposition 3, p. V.36 and Proposition 1, p. V.47].

If L F [X]/(f ), every F -algebra homomorphism L ’ Fsep is uniquely deter-

mined by the image of X, which is a root of f in Fsep . Therefore, the maps in X(L)

are in one-to-one correspondence with the roots of f in Fsep .

A ¬nite dimensional commutative F -algebra satisfying the equivalent condi-

tions above is called ´tale. From characterizations (??) or (??), it follows that ´tale

e

e

algebras remain ´tale under scalar extension.

e

Another characterization of ´tale algebras is given in Exercise ??.

e

´

Etale F -algebras and “-sets. If L is an ´tale F -algebra of dimension n,

e

Proposition (??) shows that X(L) consists of exactly n elements. The absolute

Galois group “ = Gal(Fsep /F ) acts on this set as follows:

γ

ξ =γ—¦ξ for γ ∈ “, ξ ∈ X(L).

This action is continuous since it factors through a ¬nite quotient Gal(M/F ) of “:

we may take for M any ¬nite extension of F in Fsep which contains ξ(L) for all

ξ ∈ X(L).

The construction of X(L) is functorial, since every F -algebra homomorphism

of ´tale algebras f : L1 ’ L2 induces a “-equivariant map X(f ) : X(L2 ) ’ X(L1 )

e

de¬ned by

ξ X(f ) = ξ —¦ f for ξ ∈ X(L2 ).

Therefore, writing Et F for the category of ´tale F -algebras and Sets “ for the cate-

e

gory of ¬nite sets endowed with a continuous left action of “, there is a contravariant

functor

X : Et F ’ Sets “

which associates to L ∈ Et F the “-set X(L).

282 V. ALGEBRAS OF DEGREE THREE

We now de¬ne a functor in the opposite direction. For X ∈ Sets “ , consider

the Fsep -algebra Map(X, Fsep ) of all functions X ’ Fsep . For f ∈ Map(X, Fsep )

and ξ ∈ X, it is convenient to write f, ξ for the image of ξ under f . We de¬ne a

semilinear action of “ on Map(X, Fsep ): for γ ∈ “ and f ∈ Map(X, Fsep ), the map

γ

f is de¬ned by

’1

γ

f, ξ = γ f, γ ξ for ξ ∈ X.

If γ acts trivially on X and ¬xes f, ξ for all ξ ∈ X, then γ f = f . Therefore, the

action of “ on Map(X, Fsep ) is continuous. Let Map(X, Fsep )“ be the F -algebra of

“-invariant maps. This algebra is ´tale, since by Lemma (??)

e

Map(X, Fsep )“ —F Fsep Map(X, Fsep ) Fsep — · · · — Fsep .

Every equivariant map g : X1 ’ X2 of “-sets induces an F -algebra homomor-

phism

M (g) : Map(X2 , Fsep )“ ’ Map(X1 , Fsep )“

de¬ned by

M (g)(f ), ξ = f, ξ g for f ∈ Map(X2 , Fsep )“ , ξ ∈ X1 ,

hence there is a contravariant functor

M: Sets “ ’ Et F

which maps X ∈ Sets “ to Map(X, Fsep )“ .

(18.4) Theorem. The functors X and M de¬ne an anti-equivalence of categories

Et F ≡ Sets “ .

Under this anti-equivalence, the dimension for ´tale F -algebras corresponds to the

e

cardinality for “-sets: if L ∈ Et F corresponds to X ∈ Sets “ , i.e., X X(L) and

L M(X), then

dimF L = card X.

Moreover, the direct product (resp. tensor product) of ´tale F -algebras corresponds

e

to the disjoint union (resp. direct product) of “-sets: for L1 , . . . , Lr ´tale F -

e

algebras,

X(L1 — · · · — Lr ) = X(L1 ) ··· X(Lr )

(where is the disjoint union) and

X(L1 — · · · — Lr ) = X(L1 ) — · · · — X(Lr ),

where “ acts diagonally on the right side of the last equality.

Proof : For L ∈ Et F , the canonical F -algebra homomorphism

“

¦ : L ’ Map X(L), Fsep

carries ∈ L to the map e de¬ned by

e , ξ = ξ( ) for ξ ∈ X(L).

Since card X(L) = dimF L, we have dimFsep Map X(L), Fsep = dimF L, hence

“

dimF Map X(L), Fsep = dimF L

´

§18. ETALE AND GALOIS ALGEBRAS 283

by Lemma (??). In order to prove that ¦ is an isomorphism, it therefore su¬ces

to show that ¦ is injective. Suppose ∈ L is in the kernel of ¦. By the de¬nition

of e , this means that ξ( ) = 0 for every F -algebra homomorphism L ’ Fsep . It

follows that the isomorphism LFsep Fsep — · · · — Fsep of (??) maps — 1 to 0,

hence = 0.

For X ∈ Sets “ , there is a canonical “-equivariant map

Ψ : X ’ X Map(X, Fsep )“ ,

which associates to ξ ∈ X the homomorphism eξ de¬ned by

for f ∈ Map(X, Fsep )“ .

eξ (f ) = f, ξ

Since X Map(X, Fsep )“ = X Map(X, Fsep )“sep = X Map(X, Fsep ) , the map Ψ

F

is easily checked to be bijective. The other equations are clear.

Since direct product decompositions of an ´tale F -algebra L correspond to

e

disjoint union decompositions of X(L), it follows that L is a ¬eld if and only if

X(L) is indecomposable, which means that “ acts transitively on X(L). At the

other extreme, L F — · · · — F if and only if “ acts trivially on X(L).

Traces and norms. Let L be an ´tale F -algebra of dimension n. Besides the

e

trace TL/F and the norm NL/F , we also consider the quadratic map

SL/F : L ’ F

which yields the coe¬cient of X n’2 in the generic polynomial (see (??)).

(18.5) Proposition. Let X(L) = {ξ1 , . . . , ξn }. For all ∈ L,

TL/F ( ) = ξi ( ), SL/F ( ) = ξi ( )ξj ( ), NL/F ( ) = ξ1 ( ) · · · ξn ( ).

1¤i¤n 1¤i<j¤n

Proof : It su¬ces to check these formulas after scalar extension to Fsep . We may

thus assume L = F —· · ·—F and ξi (x1 , . . . , xn ) = xi . With respect to the canonical

basis of L over F , multiplication by (x1 , . . . , xn ) is given by the diagonal matrix

with entries x1 , . . . , xn , hence the formulas are clear.

When the ´tale algebra L is ¬xed, we set T and bS for the symmetric bilinear

e

forms on L de¬ned by

(18.6) T (x, y) = TL/F (xy) and bS (x, y) = SL/F (x + y) ’ SL/F (x) ’ SL/F (y)

for all x, y ∈ L. From (??), it follows that

T (x, y) = ξi (x)ξi (y) and bS (x, y) = ξi (x)ξj (y).

1¤i¤n 1¤i=j¤n

Therefore,

(18.7) T (x, y) + bS (x, y) = TL/F (x)TL/F (y) for x, y ∈ L.

(This formula also follows readily from the general relations among the coe¬cients

of the characteristic polynomial: see (??).) By putting y = x in this equation, we

obtain:

TL/F (x2 ) + 2SL/F (x) = TL/F (x)2 for x ∈ L,

hence the quadratic form TL/F (x2 ) is singular if char F = 2 and n ≥ 2. Proposi-

tion (??) shows however that the bilinear form T is always nonsingular.

284 V. ALGEBRAS OF DEGREE THREE

Let L0 be the kernel of the trace map:

L0 = { x ∈ L | TL/F (x) = 0 }

and let S 0 : L0 ’ F be the restriction of SL/F to L0 . We write bS 0 for the polar

form of S 0 .

(18.8) Proposition. Suppose L is an ´tale F -algebra of dimension n.

e

(1) The bilinear form bS is nonsingular if and only if char F does not divide n ’ 1.

If char F divides n ’ 1, then the radical of bS is F .

(2) The bilinear form bS 0 is nonsingular if and only if char F does not divide n. If

char F divides n, then the radical of bS 0 is F .

(3) If char F = 2, the quadratic form L/F is nonsingular if and only if n ≡ 1

mod 4; the quadratic form S 0 is nonsingular if and only if n ≡ 0 mod 4.

Proof : (??) It su¬ces to prove the statements after scalar extension to Fsep . We

may thus assume that L = F — · · · — F , hence

bS (x1 , . . . , xn ), (y1 , . . . , yn ) = xi y j

1¤i=j¤n

for x1 , . . . , xn , y1 , . . . , yn ∈ F . The matrix M of bS with respect to the canonical

basis of L satis¬es:

M + 1 = (1)1¤i,j¤n .

Therefore, (M + 1)2 = n(M + 1). If char F divides n, it follows that M + 1 is

nilpotent. If char F does not divide n, the matrix n’1 (M + 1) is an idempotent of

rank 1. In either case, the characteristic polynomial of M + 1 is X n’1 (X ’ n), so

that of M is (X + 1)n’1 (X + 1) ’ n ; hence,

det M = (’1)n’1 (n ’ 1).

It follows that bS is nonsingular if and only if char F does not divide n ’ 1.

If char F divides n ’ 1, then the rank of M is n ’ 1, hence the radical of bS has

dimension 1. This radical contains F , since (??) shows that for ± ∈ F and x ∈ L,

bS (±, x) = TL/F (±)TL/F (x) ’ TL/F (±x) = (n ’ 1)±TL/F (x) = 0.

Therefore, the radical of bS is F .

(??) Equation (??) shows that bS 0 (x, y) = ’T (x, y) for all x, y ∈ L0 and that

bS (±, x) = 0 = T (±, x) for ± ∈ F and x ∈ L0 .

If char F does not divide n, then L = F • L0 ; the elements in the radical of bS 0

then lie also in the radical of T . Since T is nonsingular, it follows that bS 0 must

also be nonsingular.

If char F divides n, then F is in the radical of bS 0 . On the other hand, the ¬rst

part of the proposition shows that bS is nonsingular, hence the radical of bS 0 must

have dimension 1; this radical is therefore F .

(??) Assume char F = 2. From (??), it follows that the quadratic form SL/F is

singular if and only if n is odd and SL/F (1) = 0. Similarly, it follows from (??) that

S 0 is singular if and only if n is even and SL/F (1) = 0. Since SL/F (1) = 1 n(n ’ 1),

2

the equality SL/F (1) = 0 holds for n odd if and only if n ≡ 1 mod 4; it holds for n

even if and only if n ≡ 0 mod 4.

´

§18. ETALE AND GALOIS ALGEBRAS 285

The separability idempotent. Let L be an ´tale F -algebra. Recall from

e

(??) that we may identify X(L —F L) = X(L) — X(L): for ξ, · ∈ X(L), the

F -algebra homomorphism (ξ, ·) : L —F L ’ Fsep is de¬ned by

(18.9) (ξ, ·)(x — y) = ξ(x)·(y) for x, y ∈ L.

Theorem (??) yields a canonical isomorphism:

“

∼

L —F L ’ Map X(L) — X(L), Fsep

’ .

The characteristic function on the diagonal of X(L)—X(L) is invariant under “; the

corresponding element e ∈ L —F L is called the separability idempotent of L. This

element is indeed an idempotent since every characteristic function is idempotent.

By de¬nition, e is determined by the following condition: for all ξ, · ∈ X(L),

0 if ξ = ·,

(ξ, ·)(e) =

1 if ξ = ·.

(18.10) Proposition. Let µ : L —F L ’ L be the multiplication map. The sepa-

rability idempotent e ∈ L —F L is uniquely determined by the following conditions:

µ(e) = 1 and e(x — 1) = e(1 — x) for all x ∈ L. The map µ : L ’ e(L —F L) which

carries x ∈ L to e(x — 1) is an F -algebra isomorphism.

Proof : In view of the canonical isomorphisms

“ “

∼ ∼

L ’ Map X(L), Fsep

’ and L —F L ’ Map X(L) — X(L), Fsep

’ ,

the conditions µ(e) = 1 and e(x — 1) = e(1 — x) for all x ∈ L are equivalent to

ξ µ(e) = 1 and (ξ, ·) e(x — 1) = (ξ, ·) e(1 — x)

for all ξ, · ∈ X(L) and x ∈ L. We have

(ξ, ·) e(x — 1) = (ξ, ·)(e)ξ(x) and (ξ, ·) e(1 — x) = (ξ, ·)(e)·(x).

Therefore, the second condition holds if and only if (ξ, ·)(e) = 0 for ξ = ·.

On the other hand, ξ µ(e) = (ξ, ξ)(e), hence the ¬rst condition is equivalent

to: (ξ, ξ)(e) = 1 for all ξ ∈ X(L). This proves that the separability idempotent is

uniquely determined by the conditions of the proposition.

The map µ is injective since µ—¦µ = IdL . It is also surjective since the properties

of e imply:

e(x — y) = e(xy — 1) = µ(xy)

for all x, y ∈ L.

(18.11) Example. Let L = F [X]/(f ) for some polynomial

f = X n + an’1 X n’1 + · · · + a1 X + a0

with no repeated roots in an algebraic closure of F . Let x = X + (f ) be the image

of X in L and let

m’1

xi — xm’1’i ∈ L —F L for m = 1, . . . , n.

tm =

i=0

(In particular, t1 = 1.) The hypothesis on f implies that its derivative f is rela-

tively prime to f , hence f (x) ∈ L is invertible.

We claim that the separability idempotent of L is

e = (tn + an’1 tn’1 + · · · + a1 t1 ) f (x)’1 — 1 .

286 V. ALGEBRAS OF DEGREE THREE

Indeed, we have µ(e) = 1 since

µ(tn + an’1 tn’1 + · · · + a1 t1 ) = nxn’1 + (n ’ 1)an’1 xn’1 + · · · + a1 = f (x).

Also, tm (x — 1 ’ 1 — x) = xm — 1 ’ 1 — xm , hence

(tn + an’1 tn’1 + · · · + a1 t1 )(x — 1 ’ 1 — x) =

f (x) ’ a0 — 1 ’ 1 — f (x) ’ a0 = 0.

Therefore, e(x — 1 ’ 1 — x) = 0 and, for m = 1, . . . , n ’ 1,

e(xm — 1 ’ 1 — xm ) = e(x — 1 ’ 1 — x)tm = 0.

Since (xi )0¤i¤n’1 is a basis of L over F , it follows that e( — 1 ’ 1 — ) = 0 for all

∈ L, proving the claim.

An alternate construction of the separability idempotent is given in the follow-

ing proposition:

(18.12) Proposition. Let L be an ´tale F -algebra of dimension n = dimF L and

e

let (ui )1¤i¤n be a basis of L. Suppose (vi )1¤i¤n is the dual basis for the bilinear

form T of (??), in the sense that

T (ui , vj ) = δij (Kronecker delta) for i, j = 1, . . . , n.

n

The element e = ui — vi ∈ L — L is the separability idempotent of L.

i=1

Proof : Since (ui )1¤i¤n and (vi )1¤i¤n are dual bases, we have for x ∈ L

n n

(18.13) x= ui T (vi , x) = vi T (ui , x).

i=1 i=1

In particular, vj = ui T (vi , vj ) and uj = vi T (ui , uj ) for all j = 1, . . . , n,

i i

hence

e= ui — uj T (vi , vj ) = vi — vj T (ui , uj ).

i,j i,j

Using this last expression for e, we get for all x ∈ L:

e(x — 1) = vi x — vj T (ui , uj ).

i,j

By (??), we have vi x = uk T (vi x, vk ), hence

k

e(x — 1) = uk — vj T (ui , uj )T (vi x, vk ).

i,j,k

Since T (vi x, vk ) = TL/F (vi xvk ) = T (vi , vk x), we have

T (ui , uj )T (vi x, vk ) = T ui T (vi , vk x), uj = T (vk x, uj ),

i i

hence

e(x — 1) = uk — vj T (vk x, uj ).

j,k

Similarly, by using the expression e = ui — uj T (vi , vj ), we get for all x ∈ L:

i,j

e(1 — x) = ui — vk T (uk x, vi ).

i,k

It follows that e(x — 1) = e(1 — x) since for all ±, β = 1, . . . , n,

T (v± x, uβ ) = TL/F (v± xuβ ) = T (uβ x, v± ).

By (??) we also have for all x ∈ L and for all j = 1, . . . , n

xuj = ui T (vi , xuj ) = ui TL/F (xuj vi ),

i i

´

§18. ETALE AND GALOIS ALGEBRAS 287

hence TL/F (x) = TL/F (xui vi ). It follows that for all x ∈ L

i

T (x, 1) = TL/F (x) = TL/F xui vi = T x, ui v i ,

i i

hence i ui vi = 1 since the bilinear form T is nonsingular. This proves that

µ(e) = 1. We have thus shown that e satis¬es the conditions of (??).

18.B. Galois algebras. In this subsection, we consider ´tale F -algebras L

e

endowed with an action by a ¬nite group G of F -automorphisms. Such algebras are

called G-algebras over F . We write LG for the subalgebra of G-invariant elements:

LG = { x ∈ L | g(x) = x for all g ∈ G }.

In view of the anti-equivalence Et F ≡ Sets “ , there is a canonical isomorphism of

groups:

AutF (L) = AutSets “ X(L)

which associates to every automorphism ± of the ´tale algebra L the “-equivariant

e

±

permutation of X(L) mapping ξ ∈ X(L) to ξ = ξ —¦ ±. Therefore, an action of G

on L amounts to an action of G by “-equivariant permutations on X(L).

(18.14) Proposition. Let L be a G-algebra over F . Then, LG = F if and only if

G acts transitively on X(L).

“

∼

Proof : Because of the canonical isomorphism ¦ : L ’ Map X(L), Fsep , for x ∈

’

L the condition x ∈ LG is equivalent to: ξ —¦ g(x) = ξ(x) for all ξ ∈ X(L), g ∈ G.

Since ξ —¦g = ξ g , this observation shows that ¦ carries LG onto the set of “-invariant

maps X(L) ’ Fsep which are constant on each G-orbit of X(L). On the other hand,

¦ maps F onto the set of “-invariant maps which are constant on X(L). Therefore,

if G has only one orbit on X(L), then LG = F .

To prove the converse, it su¬ces to show that if G has at least two orbits,

then there is a nonconstant “-invariant map X(L) ’ Fsep which is constant on

each G-orbit of X(L). Since G acts by “-equivariant permutations on X(L), the

group “ acts on the G-orbits of X(L). If this action is not transitive, we may ¬nd

a disjoint union decomposition of “-sets X(L) = X1 X2 where X1 and X2 are

preserved by G. The map f : X(L) ’ Fsep de¬ned by

0 if ξ ∈ X1

f, ξ =

1 if ξ ∈ X2

is “-invariant and constant on each G-orbit of X(L).

For the rest of the proof, we may thus assume that “ acts transitively on the

G-orbits of X(L). Then, ¬xing an arbitrary element ξ0 ∈ X(L), we have

g

X(L) = { γ ξ0 | γ ∈ “, g ∈ G }.

Since G has at least two orbits in X(L), there exists ρ ∈ “ such that ρ ξ0 does not

lie in the G-orbit of ξ0 . Let a ∈ Fsep satisfy ρ(a) = a and γ(a) = a for all γ ∈ “

such that γ ξ0 belongs to the G-orbit of ξ0 . We may then de¬ne a “-invariant map

f : X(L) ’ Fsep by

g

f, γ ξ0 = γ(a) for all γ ∈ “, g ∈ G.

The map f is clearly constant on each G-orbit of X(L), but it is not constant since

f (ρ ξ0 ) = f (ξ0 ).

288 V. ALGEBRAS OF DEGREE THREE

(18.15) De¬nitions. Let L be a G-algebra over F for which the order of G equals

the dimension of L:

|G| = dimF L.

The G-algebra L is said to be Galois if LG = F . By the preceding proposition, this

condition holds if and only if G acts transitively on X(L). Since |G| = card X(L),

it then follows that the action of G is simply transitive: for all ξ, · ∈ X(L) there is

a unique g ∈ G such that · = ξ g . In particular, the action of G on L and on X(L)

is faithful.

A “-set endowed with a simply transitive action of a ¬nite group G is called a

G-torsor (or a principal homogeneous set under G). Thus, a G-algebra L is Galois

if and only if X(L) is a G-torsor. (A more general notion of torsor, allowing a

nontrivial action of “ on G, will be considered in §??.)

(18.16) Example. Let L be a Galois G-algebra over F . If L is a ¬eld, then Galois

theory shows G = AutAlg F (L). Therefore, a Galois G-algebra structure on a ¬eld L

exists if and only if the extension L/F is Galois with Galois group isomorphic to G;

∼

the G-algebra structure is then given by an isomorphism G ’ Gal(L/F ).

’

If L is not a ¬eld, then it may be a Galois G-algebra over F for various non-

isomorphic groups G. For instance, suppose L = K — K where K is a quadratic

Galois ¬eld extension of F with Galois group {Id, ±}. We may de¬ne an action of

Z/4Z on L by

(1 + 4Z)(k1 , k2 ) = ±(k2 ), k1 for k1 , k2 ∈ K.

This action gives L the structure of a Galois Z/4Z-algebra over F . On the other

hand, L also is a Galois (Z/2Z) — (Z/2Z)-algebra over F for the action:

(1 + 2Z, 0)(k1 , k2 ) = (k2 , k1 ), (0, 1 + 2Z)(k1 , k2 ) = ±(k1 ), ±(k2 )

”but not for the action

(1 + 2Z, 0)(k1 , k2 ) = ±(k1 ), k2 , (0, 1 + 2Z)(k1 , k2 ) = k1 , ±(k2 ) ,

since LG = F — F .

More generally, if M/F is a Galois extension of ¬elds with Galois group H, the

following proposition shows that one can de¬ne on M r = M —· · ·—M a structure of

Galois G-algebra over F for every group G containing H as a subgroup of index r:

(18.17) Proposition. Let G be a ¬nite group and H ‚ G a subgroup. For every

Galois H-algebra M over F there is a Galois G-algebra Ind G M over F and a

H

homomorphism π : IndG M ’ M such that π h(x) = h π(x) for all x ∈ IndG M ,

H H

h ∈ H. There is an F -algebra isomorphism:

IndG M M —···—M .

H

[G:H]

Moreover, the pair (IndG M, π) is unique in the sense that if L is another Galois G-

H

algebra over F and „ : L ’ M is a homomorphism such that „ h(x) = h „ (x) for

all x ∈ L and h ∈ H, then there is an isomorphism of G-algebras m : L ’ IndG M H

such that „ = π —¦ m.

Proof : Let

IndG M = { f ∈ Map(G, M ) | f, hg = h f, g for h ∈ H, g ∈ G },

H

´

§18. ETALE AND GALOIS ALGEBRAS 289

which is an F -subalgebra of Map(G, M ). If g1 , . . . , gr ∈ G are representatives of

the right cosets of H in G, so that

G = Hg1 ··· Hgr ,

then there is an isomorphism of F -algebras: IndG M ’ M r which carries every

H

map in IndG M to the r-tuple of its values on g1 , . . . , gr . Therefore,

H

dimF IndG M = r dimF M = |G|.

H

The algebra IndG M carries a natural G-algebra structure: for f ∈ Ind G M and

H H

g ∈ G, the map g(f ) is de¬ned by the equation:

g(f ), g = f, g g for g ∈ G.

From this de¬nition, it follows that (IndG M )G = F , proving IndG M is a Galois G-

H H

G

algebra over F . There is a homomorphism π : IndH M ’ M such that π h(x) =

h π(x) for all x ∈ L, h ∈ H, given by

π(f ) = f, 1 .

If L is a Galois G-algebra and „ : L ’ M is a homomorphism as above, we may

de¬ne an isomorphism m : L ’ IndG M by mapping ∈ L to the map m de¬ned

H

by

m , g = „ g( ) for g ∈ G.

Details are left to the reader.

It turns out that every Galois G-algebra over F has the form IndG M for some

H

Galois ¬eld extension M/F with Galois group isomorphic to H:

(18.18) Proposition. Let L be a Galois G-algebra over F and let e ∈ L be a

primitive idempotent, i.e., an idempotent which does not decompose into a sum of

nonzero idempotents. Let H ‚ G be the stabilizer subgroup of e and let M = eL.

The algebra M is a Galois H-algebra and a ¬eld, and there is an isomorphism of

G-algebras:

IndG M.

L H

Proof : Since e is primitive, the ´tale algebra M has no idempotent other than 0

e

and 1, hence (??) shows that M is a ¬eld. The action of G on L restricts to an

action of H on M . Let e1 , . . . , er be the various images of e under the action of G,

with e = e1 , say. Since each ei is a di¬erent primitive idempotent, the sum of the

ei is an idempotent in LG = F , hence e1 + · · · + er = 1 and therefore

L = e1 L — · · · — er L.

Choose g1 , . . . , gr ∈ G such that ei = gi (e); then gi (M ) = ei L, hence the ¬elds

e1 L, . . . , er L are all isomorphic to M and

dimF L = r dimF M.

On the other hand, the coset decomposition G = g1 H ··· gr H shows that

|G| = r |H|, hence

dimF M = |H|.

To complete the proof that M is a Galois H-algebra, we must show that M H = F .

r

Suppose e ∈ M H , for some ∈ L; then i=1 gi (e ) ∈ LG = F , hence

r

e gi (e ) ∈ eF.

i=1

290 V. ALGEBRAS OF DEGREE THREE

Since g1 ∈ H and egi (e) = e1 ei = 0 for i = 1, we have

r

e gi (e ) = e ,

i=1

hence e ∈ eF and M H = F .

Multiplication by e de¬nes an F -algebra homomorphism „ : L ’ M such that

„ h(x) = h π(x) for all x ∈ L, hence (??) yields a G-algebra isomorphism L

IndG M .

H

Galois algebras and torsors. Let G“Gal F denote the category of Galois G-

algebras over F , where the maps are the G-equivariant homomorphisms, and let

G“Tors “ be the category of “-sets with a G-torsor structure (for an action of G

on the right commuting with the action of “ on the left) where the maps are “-

and G-equivariant functions. As observed in (??), we have X(L) ∈ G“Tors “ for all

L ∈ G“Gal F . This construction de¬nes a contravariant functor

X : G“Gal F ’ G“Tors “ .

To obtain a functor M : G“Tors “ ’ G“Gal F , we de¬ne a G-algebra structure on the

´tale algebra Map(X, Fsep )“ for X ∈ G“Tors “ : for g ∈ G and f ∈ Map(X, Fsep ),

e

the map g(f ) : X ’ Fsep is de¬ned by

g(f ), ξ = f, ξ g for ξ ∈ X.

Since the actions of “ and G on X commute, it follows that the actions on the

algebra Map(X, Fsep ) also commute, hence the action of G restricts to an action

on Map(X, Fsep )“ . The induced action on X Map(X, Fsep )“ coincides with the

∼

action of G on X under the canonical bijection Ψ : X ’ X Map(X, Fsep )“ , hence

’

Map(X, Fsep )“ is a Galois G-algebra over F for X ∈ G“Tors “ . We let M(X) =

Map(X, Fsep )“ , with the G-algebra structure de¬ned above.

(18.19) Theorem. The functors X and M de¬ne an anti-equivalence of cate-

gories:

G“Gal F ≡ G“Tors “ .

Proof : For L ∈ Et F and X ∈ Sets “ , canonical isomorphisms are de¬ned in the

proof of (??):

“

∼ ∼

Ψ : X ’ X Map(X, Fsep )“ .

¦ : L ’ Map X(L), Fsep

’ , ’

To establish the theorem, it su¬ces to verify that ¦ and Ψ are G-equivariant if

L ∈ G“Gal F and X ∈ G“Tors “ , which is easy. (For Ψ, this was already observed

above).

The discriminant of an ´tale algebra. The Galois closure and the discrim-

e

inant of an ´tale F -algebra are de¬ned by a construction involving its associated

e

“-set. For X a “-set of n elements, let Σ(X) be the set of all permutations of a list

of the elements of X:

Σ(X) = { (ξ1 , . . . , ξn ) | ξ1 , . . . , ξn ∈ X, ξi = ξj for i = j }.

This set carries the diagonal action of “:

γ

(ξ1 , . . . , ξn ) = (γ ξ1 , . . . , γ ξn ) for γ ∈ “

and also an action of the symmetric group Sn :

(ξ1 , . . . , ξn )σ = (ξσ(1) , . . . , ξσ(n) ) for σ ∈ Sn .

´

§18. ETALE AND GALOIS ALGEBRAS 291

Clearly, Σ(X) is a torsor under Sn :

Σ(X) ∈ Sn “Tors “ ,

and the projections on the various components de¬ne “-equivariant maps

πi : Σ(X) ’ X

for i = 1, . . . , n.

Let ∆(X) be the set of orbits of Σ(X) under the action of the alternating

group An , with the induced action of “:

∆(X) = Σ(X)/An ∈ Sets “ .

This “-set has two elements.

The anti-equivalences Et F ≡ Sets “ and Sn “Gal F ≡ Sn “Tors “ yield correspond-

ing constructions for ´tale algebras. If L is an ´tale algebra of dimension n over F ,

e e

we set

“

(18.20) Σ(L) = Map Σ X(L) , Fsep ,

a Galois Sn -algebra over F with n canonical embeddings µ1 , . . . , µn : L ’ Σ(L)

de¬ned by the relation

µi ( ), (ξ1 , . . . , ξn ) = ξi ( ) for i = 1, . . . , n, ∈ L, (ξ1 , . . . , ξn ) ∈ Σ X(L) .

We also set

“

∆(L) = Map ∆ X(L) , Fsep ,

a quadratic ´tale algebra over F which may alternately be de¬ned as

e

∆(L) = Σ(L)An .

From the de¬nition of ∆ X(L) , it follows that an element γ ∈ “ acts trivially on

this set if and only if the induced permutation ξ ’ γ ξ of X(L) is even. Therefore,

the kernel of the action of “ on ∆ X(L) is the subgroup “0 ‚ “ which acts by

even permutations on X(L), and

F —F if “0 = “,

(18.21) ∆(L)

(Fsep )“0 if “0 = “.

The algebra Σ(L) is called the Galois Sn -closure of the ´tale algebra L and

e

∆(L) is called the discriminant of L.

(18.22) Example. Suppose L is a ¬eld; it is then a separable extension of degree n

of F , by (??). We relate Σ(L) to the (Galois-theoretic) Galois closure of L.

Number the elements of X(L):

X(L) = {ξ1 , . . . , ξn }

and let M be the sub¬eld of Fsep generated by ξ1 (L), . . . , ξn (L):

M = ξ1 (L) · · · ξn (L) ‚ Fsep .

Galois theory shows M is the Galois closure of each of the ¬elds ξ1 (L), . . . , ξn (L).

The action of “ on X(L) factors through an action of the Galois group Gal(M/F ).

Letting H = Gal(M/F ), we may therefore identify H with a subgroup of Sn : for

h ∈ H and i = 1, . . . , n we de¬ne h(i) ∈ {1, . . . , n} by

h

ξi = h —¦ ξi = ξh(i) .

292 V. ALGEBRAS OF DEGREE THREE

We claim that

IndSn M

Σ(L) as Sn -algebras.

H

(In particular, Σ(L) M if H = Sn ). The existence of such an isomorphism

follows from (??) if we show that there is a homomorphism „ : Σ(L) ’ M such

that „ h(f ) = h „ (f ) for all f ∈ Σ(L), h ∈ H.

“

For f ∈ Σ(L) = Map Σ X(L) , Fsep , set

„ (f ) = f, (ξ1 , . . . , ξn ) .

The right side lies in M since γ ξi = ξi for all γ ∈ Gal(Fsep /M ). For h ∈ H and

f ∈ Σ(L), we have

„ h(f ) = h(f ), (ξ1 , . . . , ξn ) = f, (ξ1 , . . . , ξn )h = f, (ξh(1) , . . . , ξh(n) ) .

On the other hand, since f is invariant under the “-action on Map Σ X(L) , Fsep ,

= f, h (ξ1 , . . . , ξn ) .

h „ (f ) = h f, (ξ1 , . . . , ξn )

Since h (ξ1 , . . . , ξn ) = (h ξ1 , . . . , h ξn ) = (ξh(1) , . . . , ξh(n) ), the claim is proved.

(18.23) Example. Suppose L = F [X]/(f ) where f is a polynomial of degree n

with no repeated roots in an algebraic closure of F . We give an explicit description

of ∆(L).

Let x = X + (f ) be the image of X in L and let x1 , . . . , xn be the roots of f

in Fsep . An F -algebra homomorphism L ’ Fsep is uniquely determined by the

image of x, which must be one of the xi . Therefore, X(L) = {ξ1 , . . . , ξn } where

ξi : L ’ Fsep maps x to xi .

If char F = 2, an element γ ∈ “ induces an even permutation of X(L) if and

only if

γ (xi ’ xj ) = (xi ’ xj ),

1¤i<j¤n 1¤i<j¤n

since

(xi ’ xj ) = ξi (x) ’ ξj (x)

1¤i<j¤n 1¤i<j¤n

and

γ

ξi (x) ’ γ ξj (x) .

γ (xi ’ xj ) =

1¤i<j¤n 1¤i<j¤n

By (??), it follows that

F [T ]/(T 2 ’ d)

∆(L)

where d = 1¤i<j¤n (xi ’ xj )2 ∈ F .

If char F = 2, the condition that γ induces an even permutation of X(L)

amounts to γ(s) = s, where

xi

s= ,

xi + x j

1¤i<j¤n

hence

F [T ]/(T 2 + T + d)

∆(L)

xi xj

where d = s2 + s = ∈ F.

1¤i<j¤n x2 +x2

i j

´

§18. ETALE AND GALOIS ALGEBRAS 293

The following proposition relates the discriminant ∆(L) to the determinant of

the trace forms on L. Recall from (??) that the bilinear form T on L is nonsin-

gular; if char F = 2, Proposition (??) shows that the quadratic form SL/F on L

is nonsingular if dimF L is even and the quadratic form S 0 on L0 = ker TL/F is

nonsingular if dimF L is odd.

(18.24) Proposition. Let L be an ´tale F -algebra of dimension n.

e

If char F = 2,

F [t]/(t2 ’ d)

∆(L)

where d ∈ F — represents the determinant of the bilinear form T .

If char F = 2,

F [t]/(t2 + t + a)

∆(L)

where a ∈ F represents the determinant of the quadratic form SL/F if n is even

1

and a + 2 (n ’ 1) represents the determinant of the quadratic form S 0 if n is odd.

Proof : Let X(L) = {ξ1 , . . . , ξn } and let “0 ‚ “ be the subgroup which acts on X(L)

by even permutations, so that ∆(L) is determined up to F -isomorphism by (??).

The idea of the proof is to ¬nd an element u ∈ Fsep satisfying the following condi-

tions:

(a) if char F = 2:

u if γ ∈ “0 ,

γ(u) =

’u if γ ∈ “ “0 ,

and u2 ∈ F — represents det T ∈ F — /F —2 .

(b) if char F = 2:

u if γ ∈ “0 ,

γ(u) =

u + 1 if γ ∈ “ “0 ,

and „˜(u) = u2 + u ∈ F represents det SL/F ∈ F/„˜(F ) if n is even, u2 +

1

u + 2 (n ’ 1) represents det S 0 ∈ F/„˜(F ) if n is odd.

The proposition readily follows, since in each case F (u) = (Fsep )“0 .

Suppose ¬rst that char F = 2. Let (ei )1¤i¤n be a basis of L over F . Consider

the matrix

M = ξi (ej ) ∈ Mn (Fsep )

1¤i,j¤n

and

u = det M ∈ Fsep .

For γ ∈ “ we have γ(u) = det γ ξi (ej ) 1¤i,j¤n . Since an even permutation of the

rows of a matrix does not change its determinant and an odd permutation changes

its sign, it follows that γ(u) = u if γ ∈ “0 and γ(u) = ’u if γ ∈ “ “0 . Moreover,

by (??) we have:

n

Mt · M = k=1 ξk (ei )ξk (ej ) 1¤i,j¤n = T (ei , ej ) ,

1¤i,j¤n

hence u2 represents det T . This completes the proof in the case where char F = 2.

294 V. ALGEBRAS OF DEGREE THREE

Suppose next that char F = 2 and n is even: n = 2m. Let (ei )1¤i¤n be a

symplectic basis of L for the bilinear form bS , so that the matrix of bS with respect

to this basis is

«

J 0

01

¬ ·

..

A= where J = 1 0 .

.

0 J

Assume moreover that e1 = 1. For i, j = 1, . . . , n, de¬ne

1¤k< ¤n ξk (ei )ξ (ej ) if i > j,

bij =

0 if i ¤ j.

Let B = (bij )1¤i,j¤n ∈ Mn (Fsep ) and let

m

u = tr(A’1 B) = i=1 b2i,2i’1 .

Under the transposition permutation of ξ1 , . . . , ξn which exchanges ξr and ξr+1

and ¬xes ξi for i = r, r + 1, the element bij is replaced by bij + ξr (ei )ξr+1 (ej ) +

ξr (ej )ξr+1 (ei ), hence u becomes u + µ where

m

µ= ξr (e2i )ξr+1 (e2i’1 ) + ξr (e2i’1 )ξr+1 (e2i ) .

i=1

Claim. µ = 1.

Since (ei )1¤i¤n is a symplectic basis of L for the bilinear form bS , it follows

from (??) that TL/F (e2 ) = 1 and TL/F (ei ) = 0 for i = 2. The dual basis for the

bilinear form T is then (fi )1¤i¤n where

f1 = e1 + e2 , f2 = e1 , and f2i’1 = e2i , f2i = e2i’1 for i = 2, . . . , m.

Proposition (??) shows that n ei — fi ∈ L — L is the separability idempotent

i=1

of L. From the de¬nition of this idempotent, it follows that

n n

(ξr , ξr+1 ) ei — f i = i=1 ξr (ei )ξr+1 (fi ) = 0.

i=1

(See (??) for the notation.) On the other hand, the formulas above for f1 , . . . , fn

show:

n m

i=1 ei — fi = e1 — e1 + i=1 (e2i — e2i’1 + e2i’1 — e2i ).

Since e1 = 1, the claim follows.

Since the symmetric group is generated by the transpositions (r, r + 1), the

claim shows that u is transformed into u + 1 by any odd permutation of ξ1 , . . . , ξn

and is ¬xed by any even permutation. Therefore,

u if γ ∈ “0 ,

γ(u) =

u + 1 if γ ∈ “ “0 .

We proceed to show that u2 + u represents det SL/F ∈ F/„˜(F ).

Let C = 1¤k< ¤n ξk (ei )ξ (ej ) 1¤i,j¤n ∈ Mn (Fsep ) and let

D = C + B + Bt.

We have D = (dij )1¤i,j¤n where

±

0 if i > j,

dij = ξ (e )ξ (ei ) = SL/F (ei ) if i = j,

1¤k< ¤n k i

1¤k< ¤n ξk (ei )ξ (ej ) + ξk (ej )ξ (ei ) = bS (ei , ej ) if i < j,

´

§18. ETALE AND GALOIS ALGEBRAS 295

hence D ∈ Mn (F ) is a matrix of the quadratic form SL/F with respect to the basis

(ei )1¤i¤n and D + Dt = A. Therefore, s2 (A’1 D) ∈ F represents the determinant

of SL/F . Since D = C + B + B t , Lemma (??) yields

s2 (A’1 D) = s2 (A’1 C) + „˜ tr(A’1 B) = s2 (A’1 C) + u2 + u.

To complete the proof, it su¬ces to show s2 (A’1 C) = 0.

Let M = ξi (ej ) 1¤i,j¤n ∈ Mn (Fsep ) and let V = (vij )1¤i,j¤n where

0 if i ≥ j,

vij =

1 if i < j.

The matrix M is invertible, since M t · M = T (ei , ej ) 1¤i,j¤n and T is nonsingular.

Moreover, C = M t · V · M , hence A = C + C t = M t (V + V t )M and therefore

s2 (A’1 C) = s2 (V + V t )’1 V .

Observe that (V + V t + 1)2 = 0, hence (V + V t )’1 = V + V t . It follows that

s2 (V + V t )’1 V = s2 (V + V t )V = s2 (V + V t + 1)V + V .

Using the relations between coe¬cients of characteristic polynomials (see (??)), we

may expand the right-hand expression to obtain:

s2 (V + V t )’1 V = s2 (V + V t + 1)V + s2 (V ) +

tr (V + V t + 1)V tr(V ) + tr (V + V t + 1)V 2 .

Since V + V t + 1 has rank 1, we have s2 (V + V t + 1)V = 0. Since V is nilpotent,

we have tr(V ) = s2 (V ) = 0. A computation shows that tr (V + V t + 1)V 2 = 0,

hence s2 (V + V t )’1 V = 0 and the proof is complete in the case where char F = 2

and n is even.

Assume ¬nally that char F = 2 and n is odd: n = 2m + 1. Let (ei )1¤i¤n’1

be a symplectic basis of L0 for the bilinear form bS 0 . We again denote by A

the matrix of the bilinear form with respect to this basis and de¬ne a matrix

B = (bij )1¤i,j¤n’1 ∈ Mn’1 (Fsep ) by

1¤k< ¤n ξk (ei )ξ (ej ) if i > j,

bij =

0 if i ¤ j.

Let

m

u = tr(A’1 B) = i=1 b2i,2i’1 .

In order to extend (ei )1¤i¤n’1 to a basis (ei )1¤i¤n of L, de¬ne en = 1. Since

bS 0 (x, y) = T (x, y) for all x, y ∈ L0 , by (??), the dual basis (fi )1¤i¤n for the

bilinear form T is given by

f2i’1 = e2i , f2i = e2i’1 for i = 1, . . . , m, fn = en = 1.

Proposition (??) shows that the separability idempotent of L is

m

e= i=1 (e2i — e2i’1 + e2i’1 — e2i ) + en — en

hence for r = 1, . . . , n ’ 2,

m

ξr (e2i )ξr+1 (e2i’1 ) + ξr (e2i’1 )ξr+1 (e2i ) = ξr (en )ξr+1 (en ) = 1.

i=1

296 V. ALGEBRAS OF DEGREE THREE

The same argument as in the preceding case then shows

u if γ ∈ “0 ,

γ(u) =

u + 1 if γ ∈ “ “0 .

Mimicking the preceding case, we let

C= 1¤k< ¤n ξk (ei )ξ (ej ) ∈ Mn’1 (Fsep )

1¤i,j¤n’1

and

D = C + B + B t = (dij )1¤i,j¤n’1 ,

where

±

0 if i > j,

dij = S 0 (ei ) if i = j,

bS 0 (ei , ej ) if i < j.

The element s2 (A’1 D) represents the determinant det S 0 ∈ F/„˜(F ) and

s2 (A’1 D) = s2 (A’1 C) + „˜ tr(A’1 B) = s2 (A’1 C) + u2 + u,

hence it su¬ces to show s2 (A’1 C) = 1 (n ’ 1) to complete the proof.

2

For j = 1, . . . , n’1 we have ej ∈ L0 , hence ξn (ej ) = n’1 ξ (ej ), and therefore

0

=1

n’1 n’1

cij = 1¤k< ¤n’1 ξk (ei )ξ (ej ) + k=1 ξk (ei ) ξ (ej )

=1

= n’1≥k≥ ≥1 ξk (ei )ξ (ej ).

The matrix M = ξi (ej ) ∈ Mn’1 (Fsep ) is invertible since

1¤i,j¤n’1

M t · M = T (ei , ej ) = bS 0 (ei , ej )

1¤i,j¤n’1 1¤i,j¤n’1

and bS 0 is nonsingular. Moreover, C = M t · W · M where W = (wij )1¤i,j¤n’1 is

de¬ned by

1 if i ≥ j,

wij =

0 if i < j,

hence s2 (A’1 C) = s2 (W + W t )’1 W . Computations similar to those of the

preceding case show that s2 (W + W t )’1 W = 1 (n ’ 1).

2

18.C. Cubic ´tale algebras. Cubic ´tale algebras, i.e., ´tale algebras of di-

e e e

mension 3, have special features with respect to the Galois S3 -closure and discrim-

inant: if L is such an algebra, we establish below canonical isomorphisms

L —F L L — Σ(L) and Σ(L) L — ∆(L).

Moreover, we show that if F is in¬nite of characteristic di¬erent from 3, these

algebras have the form F [X]/(X 3 ’ 3X + t) for some t ∈ F , and we set up an exact

sequence relating the square class group of L to the square class group of F .

´

§18. ETALE AND GALOIS ALGEBRAS 297

The Galois closure and the discriminant. Let L be a cubic ´tale F - e

algebra. Let Σ(L) be the Galois S3 -closure of L and let e ∈ L—L be the separability

idempotent of L.

(18.25) Proposition. There are canonical F -algebra isomorphisms

∼ ∼

˜ : (1 ’ e) · (L — L) ’ Σ(L)

’ ˜ : L — L ’ L — Σ(L)

’

and

related by ˜(x — y) = xy, ˜ (1 ’ e) · (x — y) for x, y ∈ L.

Proof : Consider the disjoint union decomposition of “-sets

X(L — L) = X(L) — X(L) = D(L) E(L)

where D(L) is the diagonal of X(L) — X(L) and E(L) is its complement. By

de¬nition, e corresponds to the characteristic function on D(L) under the canonical

“

isomorphism L — L Map X(L — L), Fsep , hence

E(L) = { ξ ∈ X(L — L) | ξ(1 ’ e) = 1 }.

Therefore, we may identify

X (1 ’ e) · (L — L) = E(L),

“

hence also (1 ’ e) · (L — L) = Map E(L), Fsep . On the other hand, there is a

canonical bijection

∼

Σ X(L) ’ E(L)

’

which maps (ξi , ξj , ξk ) ∈ Σ X(L) to (ξi , ξj ) ∈ E(L). Under the anti-equivalence

Et F ≡ Sets “ , this bijection induces an isomorphism

∼

˜ : (1 ’ e) · (L — L) ’ Σ(L).

’

By decomposing L — L = e · (L — L) • (1 ’ e) · (L — L) , and combining ˜ with

∼

the isomorphism µ’1 : e · (L — L) ’ L of (??) which maps e · (x — y) = e · (xy — 1)

’

to xy, we obtain the isomorphism

∼

˜ : L — L ’ L — Σ(L).

’

∼

Note that there are actually three canonical bijections Σ X(L) ’ E(L), since

’

(ξi , ξj , ξk ) ∈ Σ X(L) may alternately be mapped to (ξi , ξk ) or (ξj , ξk ) instead of

∼

(ξi , ξj ); therefore, there are three canonical isomorphisms (1 ’ e) · (L — L) ’ Σ(L)

’

∼

and L — L ’ L — Σ(L).

’

In view of the proposition above, there is an S3 -algebra structure on (1 ’ e) ·

(L — L) and there are three embeddings 1 , 2 , 3 : L ’ (1 ’ e) · (L — L), which we

now describe: for ∈ L, we set

1( ) = (1 ’ e) · ( — 1), 2( ) = (1 ’ e) · (1 — )

and

(18.26) 3( ) = (1 ’ e) · TL/F ( )1 — 1 ’ — 1 ’ 1 — .

These isomorphisms correspond to the maps E(L) ’ X(L) which carry an

element (ξi , ξj ) ∈ E(L) respectively to ξi , ξj and to the element ξk such that

X(L) = {ξi , ξj , ξk }. Indeed, for ∈ L,

(ξi , ξj )[(1 ’ e) · ( — 1)] = ξi ( ), (ξi , ξj )[(1 ’ e) · (1 — )] = ξj ( )

298 V. ALGEBRAS OF DEGREE THREE

and

(ξi , ξj ) (1 ’ e) · TL/F ( )1 — 1 ’ — 1 ’ 1 — = TL/F ( ) ’ ξi ( ) ’ ξj ( ) = ξk ( ).

An action of S3 on (1 ’ e) · (L — L) is de¬ned by permuting the three copies of L

in (1 ’ e) · (L — L), so that σ —¦ i = σ(i) for σ ∈ S3 .

We now turn to the discriminant algebra ∆(L):

(18.27) Proposition. Let L be a cubic ´tale algebra. The canonical embeddings

e

µ1 , µ2 , µ3 : L ’ Σ(L) de¬ne isomorphisms:

∼

µ1 , µ2 , µ3 : ∆(L) —F L ’ Σ(L).

’

Proof : Consider the transpositions in S3 :

„1 = (2, 3) „2 = (1, 3) „3 = (1, 2)

and the subgroups of order 2:

Hi = {Id, „i } ‚ S3 for i = 1, 2, 3.

For i = 1, 2, 3, the canonical map

Σ X(L) ’ Σ X(L) /A3 — Σ X(L) /Hi

which carries ζ ∈ Σ X(L) to the pair (ζ A3 , ζ Hi ) consisting of its orbits under A3

and under Hi is a “-equivariant bijection, since ζ A3 © ζ Hi = {ζ}. Moreover, pro-

jection on the i-th component πi : Σ X(L) ’ X(L) factors through Σ X(L) /Hi ;

we thus get three canonical “-equivariant bijections:

∼

πi : Σ X(L) ’ ∆ X(L) — X(L).

’

Under the anti-equivalence Et F ≡ Sets “ , these bijections yield the required isomor-

phisms µi for i = 1, 2, 3.

Combining (??) and (??), we get:

(18.28) Corollary. For every cubic ´tale F -algebra L, there are canonical F -

e

algebra isomorphisms:

(Id — µi ’1 ) —¦ ˜: L —F L ’ L — ∆(L) —F L

∼

’ for i = 1, 2, 3.

The isomorphism (Id — µ2 ’1 ) —¦ ˜ is L-linear for the action of L on L —F L and on

∆(L) —F L by multiplication on the right factor.

Proof : The ¬rst assertion follows from (??) and (??). A computation shows that

˜(1 — ) maps (ξ1 , ξ2 , ξ3 ) ∈ Σ X(L) to ξ2 ( ), for all ∈ L. Similarly, µ2 (1 — )

maps (ξ1 , ξ2 , ξ3 ) to ξ2 ( ), hence µ2 ’1 —¦ ˜ is L-linear.

We conclude with two cases where the discriminant algebra can be explicitly

calculated:

(18.29) Proposition. For every quadratic ´tale F -algebra K,

e

∆(F — K) K.

Proof : The projection on the ¬rst component ξ : F — K ’ F is an element of

X(F — K) which is invariant under the action of “. If X(K) = {·, ζ}, then X(F —

K) = {ξ, ·, ζ}, and the map which carries · to (ξ, ·, ζ)A3 and ζ to (ξ, ζ, ·)A3 de¬nes

∼

an isomorphism of “-sets X(K) ’ ∆ X(F —K) . The isomorphism ∆(F —K) K

’

follows from the anti-equivalence Et F ≡ Sets “ .

´

§18. ETALE AND GALOIS ALGEBRAS 299

(18.30) Proposition. A cubic ´tale F -algebra L can be given an action of the

e

alternating group A3 which turns it into a Galois A3 -algebra over F if and only if

∆(L) F — F .

Proof : Suppose ∆(L) F — F ; then Proposition (??) yields an isomorphism

Σ(L) L — L. The action of A3 on Σ(L) preserves each term and therefore induces

an action on each of them. The induced actions are not the same, but each of them

de¬nes a Galois A3 -algebra structure on L since Σ(L) is a Galois S3 -algebra.

Conversely, suppose L has a Galois A3 -algebra structure; we have to show that

“ acts by even permutations on X(L). By way of contradiction, suppose γ ∈ “

induces an odd permutation on X(L): we may assume X(L) = {ξ1 , ξ2 , ξ3 } and

γ

ξ1 = ξ1 , γ ξ2 = ξ3 , γ ξ3 = ξ2 . Since L is a Galois A3 -algebra, X(L) is an A3 -torsor;

we may therefore ¬nd σ ∈ A3 such that

σ σ σ

ξ1 = ξ 2 , ξ2 = ξ 3 , ξ3 = ξ 1 ;

then γ (ξ1 ) = ξ3 whereas (γ ξ1 )σ = ξ2 , a contradiction. Therefore, “ acts on X(L)

σ

by even permutations, hence ∆(L) F — F .

Reduced equations. Let L be a cubic ´tale F -algebra. As a ¬rst step in

e

¬nding a reduced form for L, we relate the quadratic form S 0 on the subspace

L0 = { x ∈ L | TL/F (x) = 0 } and the bilinear form T on L to the discriminant

algebra ∆(L). Proposition (??) shows that S 0 is nonsingular if char F = 3; the

bilinear form T is nonsingular in every characteristic, since L is ´tale.

e

(18.31) Lemma. (1) The quadratic form S 0 is isometric to the quadratic form Q

on ∆(L) de¬ned by

Q(x) = N∆(L)/F (x) ’ T∆(L)/F (x)2 for x ∈ ∆(L).

(2) Suppose char F = 2 and let δ ∈ F — be such that ∆(L) F [t]/(t2 ’ δ). The

bilinear form T on L has a diagonalization

T 1, 2, 2δ .

Proof : (??) By a theorem of Springer (see Scharlau [?, Corollary 2.5.4], or Baeza

[?, p. 119] if char F = 2), it su¬ces to check that S 0 and Q are isometric over an

odd-degree scalar extension of F . If L is a ¬eld, we may therefore extend scalars to

L; then L is replaced by L — L, which is isomorphic to L — ∆(L) — L , by (??). In

all cases, we may thus assume L F — K, where K is a quadratic ´tale F -algebra.

e

Proposition (??) shows that we may identify K = ∆(L).

The generic polynomial of (±, x) ∈ F — K = L is

(X ’ ±) X 2 ’ TK/F (x)X + NK/F (x) =

X 3 ’ TL/F (±, x)X 2 + SL/K (±, x)X ’ NL/F (±, x),

hence

TL/F (±, x) = ± + TK/F (x) and SL/F (±, x) = ±TK/F (x) + NK/F (x).

Therefore, the map which carries x ∈ K = ∆(L) to ’TK/F (x), x ∈ L0 is an

∼

isometry ∆(L), Q ’ (L0 , S 0 ).

’

(??) As in (??), we may reduce to the case where L = F — K, with K

∆(L) F [t]/(t2 ’ δ). Let t be the image of t in K. A computation shows that

T (x1 , x2 + x3 t), (y1 , y2 + y3 t) = x1 y1 + 2(x2 y2 + δx3 y3 ),

300 V. ALGEBRAS OF DEGREE THREE

hence T has a diagonalization 1, 2, 2δ with respect to the basis (1, 0), (0, 1), (0, t) .

(18.32) Proposition. Every cubic ´tale F -algebra L is isomorphic to an algebra

e

of the form F [X]/(f ) for some polynomial f , unless F = F2 and L F — F — F .

If char F = 3 and F is in¬nite, the polynomial f may be chosen of the form

f = X 3 ’ 3X + a for some a ∈ F , a = ±2;

then ∆(L) F [t]/ t2 +3(a2 ’4) if char F = 2 and ∆(L) F [t]/(t2 +t+1+a’2) if

char F = 2. (Note that X 3 ’ 3X ± 2 = (X 1)2 (X ± 2), hence F [X]/(X 3 ’ 3X ± 2)

is not ´tale.)

e

If char F = 3 (and card F is arbitrary), the polynomial f may be chosen of the

form f = X 3 ’ b for some b ∈ F — if and only if ∆(L) F [t]/(t2 + t + 1).

If char F = 3, let δ ∈ F — be such that ∆(L) F [t]/(t2 ’ δ); then f may be

chosen of the form

f = X 3 ’ δX + a for some a ∈ F .

Proof : If L is a ¬eld, then it contains a primitive element x (see for instance

Bourbaki [?, p. V.39]); we then have L F [X]/(f ) where f is the minimal poly-

nomial of x. Similarly, if L F — K for some quadratic ¬eld extension K/F ,

then L F [X]/(Xg) where g is the minimal polynomial of any primitive element

of K. If L F — F — F and F contains at least three distinct elements a, b, c,

then L F [X]/(f ) where f = (X ’ a)(X ’ b)(X ’ c). This completes the proof

of the ¬rst assertion. We now show that, under suitable hypotheses, the primitive

element x may be chosen in such a way that its minimal polynomial takes a special

form.

Suppose ¬rst that char F = 3. The lemma shows that S 0 represents ’3, since

Q(1) = N∆(L)/F (1) ’ T∆(L)/F (1)2 = ’3.

Let x ∈ L0 be such that S 0 (x) = ’3. Since F ‚ L0 , the element x is a primitive

element if L is a ¬eld, and its minimal polynomial coincides with its generic poly-

nomial, which has the form X 3 ’ 3X + a for some a ∈ F . If L F — F — F , the

nonprimitive elements in L0 have the form (x1 , x2 , x3 ) where x1 + x2 + x3 = 0 and

two of the xi are equal. The conic S 0 (X) = ’3 is nondegenerate by (??). Therefore

it has only a ¬nite number of intersection points with the lines x1 = x2 , x1 = x3

and x2 = x3 . If F is in¬nite we may therefore ¬nd a primitive element x such that

S 0 (x) = ’3. Similarly, if L F — K where K is a ¬eld, the nonprimitive elements

in L0 have the form (x1 , x2 ) where x1 , x2 ∈ F and x1 + 2x2 = 0. Again, the conic

S 0 (X) = ’3 has only a ¬nite number of intersection points with this line, hence

we may ¬nd a primitive element x such that S 0 (x) = ’3 if F is in¬nite.

Example (??) shows how to compute ∆(L) for L = F [X]/(f ). If x1 , x2 , x3 are

the roots of f in an algebraic closure of F , we have

F [t]/(t2 ’ d) with d = (x1 ’ x2 )2 (x1 ’ x3 )2 (x2 ’ x3 )2

∆(L) if char F = 2,

and

x1 x2 x1 x3 x2 x3

F [t]/(t2 + t + d) with d =

∆(L) 2 + x2 + x 2 + x2 + x 2 if char F = 2.

x2+ x2

1 1 3 2 3

If f = X 3 +pX +q, then x1 +x2 +x3 = 0, x1 x2 +x1 x3 +x2 x3 = p and x1 x2 x3 = ’q,

and a computation shows that

(x1 ’ x2 )2 (x1 ’ x3 )2 (x2 ’ x3 )2 = ’4p3 ’ 27q 2 .

´

§18. ETALE AND GALOIS ALGEBRAS 301

Moreover, if char F = 2 we have x2 + x2 + x2 = 0, hence

1 2 3

x3 x3 + x 3 x3 + x 3 x3 p3 + q 2

x1 x2 x1 x3 x2 x3 12 13 23

+2 +2 = = .

x2 + x 2 x1 + x 2 x2 + x 2 x2 x2 x2 q2

1 2 3 3 123

Thus,

F [t]/(t2 + 4p3 + 27q 2 ) if char F = 2

∆(L)

and

F [t]/(t2 + t + 1 + p3 q ’2 )

∆(L) if char F = 2.

F [t]/ t2 + 3(a2 ’ 4) if f = X 3 ’ 3X + a and

In particular, we have ∆(L)

char F = 2, 3, and ∆(L) F [t]/(t2 + t + 1) if f = X 3 ’ b and char F = 3.

F [t]/(t2 + t + 1), then the form Q on ∆(L) de¬ned in

Conversely, if ∆(L)

(??) is isotropic: for j = t + (t2 + t + 1) ∈ ∆(L) we have T∆(L)/F (j) = ’1 and

N∆(L)/F (j) = 1, hence Q(j) = 0. Lemma (??) then shows that there is a nonzero

element x ∈ L0 such that S 0 (x) = 0. An inspection of the cases where L is not a

F [X]/(X 3 ’ b) where

¬eld shows that x is primitive in all cases. Therefore, L

b = x3 = NL/F (x).

Suppose ¬nally that char F = 3, and let δ ∈ F — be such that ∆(L) F [t]/(t2 ’

δ). The element d = t + (t2 ’ δ) ∈ ∆(L) then satis¬es

N∆(L)/F (d) ’ T∆(L)/F (d)2 = ’δ,

hence (??) shows that there exists x ∈ L0 such that S 0 (x) = ’δ. This element may

be chosen primitive, and its minimal polynomial then has the form

X 3 ’ δX + a for some a ∈ F .

A careful inspection of the argument in the proof above shows that if char F = 3

and L F — F — F one may ¬nd a ∈ F such that L F [X]/(X 3 ’ 3X + a) as

soon as card F ≥ 8. The same conclusion holds if L F — K for some quadratic

¬eld extension K when card F ≥ 4.

The group of square classes. Let L be a cubic ´tale F -algebra. The inclu-

e

sion F ’ L and the norm map NL/F : L ’ F induce maps on the square class

groups:

i : F — /F —2 ’ L— /L—2 , N : L— /L—2 ’ F — /F —2 .

Since NL/F (x) = x3 for all x ∈ F , the composition N —¦ i is the identity on F — /F —2 :

N —¦ i = IdF — /F —2 .

In order to relate L— /L—2 and F — /F —2 by an exact sequence, we de¬ne a map

#

: L ’ L as follows: for ∈ L we set

# 2

(18.33) = ’ TL/F ( ) + SL/F ( ) ∈ L,

so that

#

’ NL/F ( ) = 0.

In particular, for ∈ L— we have # = NL/F ( ) ’1 , hence # de¬nes an endomor-

phism of L— . We also put # for the induced endomorphism of L— /L—2 .

302 V. ALGEBRAS OF DEGREE THREE

(18.34) Proposition. The following sequence is exact:

#

i N

1 ’ F — /F —2 ’ L— /L—2 ’ L— /L—2 ’ F — /F —2 ’ 1.

’ ’ ’

Moreover, for x ∈ F — /F —2 and y ∈ L— /L—2 ,

(y # )# = i —¦ N (y) · y.

x = N —¦ i(x) and

Proof : It was observed above that N —¦ i is the identity on F — /F —2 . Therefore, i

is injective and N is surjective. For ∈ L— we have # = NL/F ( ) ’1 , hence

y # = i —¦ N (y) · y for y ∈ L— /L—2 .

(18.35)

Taking the image of each side under N , we obtain N (y # ) = 1, since N —¦ i —¦ N (y) =

N (y). Therefore, substituting y # for y in (??) we get (y # )# = y # = i —¦ N (y) · y.

In particular, if y # = 1 we have y = i —¦ N (y), and if N (y) = 1 we have y = y # .

Therefore, the kernel of # is in the image of i, and the kernel of N is in the image

of # . To complete the proof, observe that putting y = i(x) in (??) yields i(x)# = 1

for x ∈ F — /F —2 , since N —¦ i(x) = x.

§19. Central Simple Algebras of Degree Three

In this section, we turn to central simple algebras of degree 3. We ¬rst prove

Wedderburn™s theorem which shows that these algebras are cyclic, and we next

discuss their involutions of unitary type. It turns out that involutions of unitary

type on a given central simple algebra of degree 3 are classi¬ed up to conjugation

by a 3-fold P¬ster form. In the ¬nal subsection, we relate this invariant to cubic

´tale subalgebras and prove a theorem of Albert on the existence of certain cubic

e

´tale subalgebras.

e

19.A. Cyclic algebras. To simplify the notation, we set C3 = Z/3Z and

ρ = 1 + 3Z ∈ C3 .

Given a Galois C3 -algebra L over F and an element a ∈ F — , the cyclic algebra

(L, a) is de¬ned as follows:

(L, a) = L • Lz • Lz 2

where z is subject to the relations:

z3 = a

z = ρ( )z,

for all ∈ L.

(19.1) Example. Let L = F — F — F with the C3 -structure de¬ned by

ρ(x1 , x2 , x3 ) = (x3 , x1 , x2 ) for (x1 , x2 , x3 ) ∈ L.