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