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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


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