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contain Symd(»m A, γ). Again, it su¬ces to prove the result in the split case.
We may thus assume A = EndF (V ). From the description of ψm and • in this
case, it follows that the image of • —¦ ψm consists of the 1-dimensional spaces in
m m
V— V spanned by elements of the form (v1 § · · · § vm ) — (v1 § · · · § vm ),
with v1 , . . . , vm ∈ V . By (??), hyperplanes whose intersection with Symd(»m A, γ)
coincides with ker Trpγ correspond to quadratic pairs on »m A with involution γ,
m
hence to quadratic forms on V whose polar is the canonical pairing, up to a
scalar factor. Those hyperplanes which contain the image of • —¦ ψm correspond to
nonsingular quadratic forms which vanish on decomposable elements v1 § · · · § vm ,
and Proposition (??) shows that there is one and only one such quadratic form up
to a scalar factor.
§10. THE DISCRIMINANT ALGEBRA 123


(10.18) De¬nition. Let A be a central simple algebra of degree n = 2m over a
¬eld F of characteristic 2. By (??), Proposition (??) de¬nes a unique quadratic
pair (γ, f ) on »m A, which we call the canonical quadratic pair. The proof of (??)
shows that in the case where A = EndF (V ) this quadratic pair is associated with
m
the canonical map q on V de¬ned in (??). Since A may be split by a scalar
extension in which F is algebraically closed, and since the discriminant of the
canonical map q is trivial, by (??), it follows that disc(γ, f ) = 0.
If char F = 2, the canonical involution γ on »m A is orthogonal if and only if m
1
is even. Letting f be the restriction of 2 TrdA to Sym(»m A, γ), we also call (γ, f )
the canonical quadratic pair in this case. Its discriminant is trivial, as observed
in (??).
10.D. Induced involutions on »-powers. In this section, ι is an automor-
phism of the base ¬eld F such that ι2 = IdF (possibly ι = IdF ). Let V be a (¬nite
dimensional) vector space over F . Every hermitian15 form h on V with respect to ι
induces for every integer k a hermitian form h—k on V —k such that
h—k (x1 — · · · — xk , y1 — · · · — yk ) = h(x1 , y1 ) · · · h(xk , yk )
for x1 , . . . , xk , y1 , . . . , yk ∈ V . The corresponding linear map
h—k : V —k ’ ι (V —k )—
ˆ
(see (??)) is (h)—k under the canonical identi¬cation ι (V —k )— = (ι V — )—k , hence
h—k is nonsingular if h is nonsingular. Moreover, the adjoint involution σh—k on
EndF (V —k ) = EndF (V )—k is the tensor product of k copies of σh :
σh—k = (σh )—k .
k
The hermitian form h also induces a hermitian form h§k on V such that
h§k (x1 § · · · § xk , y1 § · · · § yk ) = det h(xi , yj ) 1¤i,j¤k
for x1 , . . . , xk , y1 , . . . , yk ∈ V . The corresponding linear map
k k
V ’ ι( V )—
h§k :
is k h under the canonical isomorphism k (ι V — ) ’ ι ( k
ˆ ∼
V )— which maps ι •1 §

· · · § ι •k to ι ψ where ψ ∈ ( k V )— is de¬ned by
ψ(x1 § · · · § xk ) = det •i (xj ) ,
1¤i,j¤k

for •1 , . . . , •k ∈ V — and x1 , . . . , xk ∈ V . Therefore, h§k is nonsingular if h is
nonsingular.
k
We will describe the adjoint involution σh§k on EndF ( V ) in a way which
generalizes to the »k -th power of an arbitrary central simple F -algebra with invo-
lution.
k
We ¬rst observe that if : V —k ’ V is the canonical epimorphism and
—k —k
sk : V ’V is the endomorphism considered in §??:
sk (v1 — · · · — vk ) = sgn(π)vπ’1 (1) — · · · — vπ’1 (k) ,
π∈Sk

then for all u, v ∈ V —k we have
h—k sk (u), v = h§k (u), (v) = h—k u, sk (v) .
(10.19)
15 By convention, a hermitian form with respect to IdF is a symmetric bilinear form.
124 II. INVARIANTS OF INVOLUTIONS

—k
In particular, it follows that σh (sk ) = sk .
(10.20) De¬nition. Let A be a central simple F -algebra with an involution σ
such that σ(x) = ι(x) for all x ∈ F . Recall from (??) that for k = 2, . . . , deg A,
»k A = EndA—k (A—k sk ).
According to (??), every f ∈ »k A has the form f = ρ(usk ) for some u ∈ A—k , i.e.,
there exists u ∈ A—k such that
xf = xusk for x ∈ A—k sk .
We then de¬ne σ §k (f ) = ρ σ —k (u)sk , i.e.,
§k
xσ (f )
= xσ —k (u)sk for x ∈ A—k sk .
To check that the de¬nition of σ §k (f ) does not depend on the choice of u, observe
¬rst that if f = ρ(usk ) = ρ(u sk ), then
sf = sk usk = sk u sk .
k
By applying σ —k to both sides of this equation, and taking into account the fact
that sk is symmetric under σ —k (see (??)), we obtain
sk σ —k (u)sk = sk σ —k (u )sk .
Since every x ∈ A—k sk has the form x = ysk for some y ∈ A—k , it follows that
xσ —k (u)sk = ysk σ —k (u)sk = ysk σ —k (u )sk = xσ —k (u )sk .
This shows that σ §k (f ) is well-de¬ned. Since σ(x) = ι(x) for all x ∈ F , it is easily
veri¬ed that σ §k also restricts to ι on F .
1
A = A and we set σ §1 = σ.
For k = 1, we have
(10.21) Proposition. If A = EndF (V ) and σ = σh is the adjoint involution
with respect to some nonsingular hermitian form h on V , then under the canonical
isomorphism »k A = EndF ( k V ), the involution σ §k is the adjoint involution with
k
respect to the hermitian form h§k on V.
Proof : Recall the canonical isomorphism of (??):
k k
A—k sk = HomF ( V, V —k ), hence »k A = EndF ( V ).
k
For f = ρ(usk ) ∈ EndA—k (A—k sk ), the corresponding endomorphism • of V is
de¬ned by
•(x1 § · · · § xk ) = —¦ u —¦ sk (x1 — · · · — xk )
or
• —¦ = —¦ u —¦ sk
k
where : V —k ’ V is the canonical epimorphism. In order to prove the propo-
sition, it su¬ces, therefore, to show:
h§k (x), —¦ u —¦ sk (y) = h§k —¦ σ —k (u) —¦ sk (x), (y)
for all x, y ∈ V —k . From (??) we have
h§k (x), —¦ u —¦ sk (y) = h—k sk (x), u —¦ sk (y)
and
h§k —¦ σ —k (u) —¦ sk (x), (y) = h—k σ —k (u) —¦ sk (x), sk (y) .
§10. THE DISCRIMINANT ALGEBRA 125


The proposition then follows from the fact that σ —k is the adjoint involution with
respect to h—k .
The next proposition is more speci¬cally concerned with symmetric bilinear
§m
forms b. In the case where dim V = n = 2m, we compare the involution σb with
m
the canonical involution γ on EndF ( V ).
(10.22) Proposition. Let b be a nonsingular symmetric, nonalternating, bilinear
form on an F -vector space V of dimension n = 2m. Let (e1 , . . . , en ) be an orthog-
n
onal basis of V and let e = e1 § · · · § en ∈ V ; let also
n
m
δ = (’1) b(ei , ei ),
i=1
m
so that disc b = δ · F —2 . There is a map u ∈ EndF ( V ) such that
b§m u(x), y e = x § y = (’1)m b§m x, u(y) e
(10.23)
m
for all x, y ∈ V , and
u2 = δ ’1 · Id§m V .
(10.24)
If σ = σb is the adjoint involution with respect to b, then the involution σ §m on
m
EndF ( V ) is related to the canonical involution γ by
σ §m = Int(u) —¦ γ.
In particular, the involutions σ §m and γ commute.
Moreover, if char F = 2 and m ≥ 2, the map u is a similitude of the canonical
m n
V of (??) with multiplier δ ’1 , i.e.,
quadratic map q : V’
q u(x) = δ ’1 q(x)
m
for all x ∈ V.
Proof : Let ai = b(ei , ei ) ∈ F — for i = 1, . . . , n. As in (??), we set
m
e S = e i1 § · · · § e im ∈ V and let a S = a i1 · · · a im
for S = {i1 , . . . , im } ‚ {1, . . . , n} with i1 < · · · < im . If S = T , the matrix
b(ei , ej ) (i,j)∈S—T has at least one row and one column of 0™s, namely the row corre-
sponding to any index in S T and the column corresponding to any index in T S.
Therefore, b§m (eS , eT ) = 0. On the other hand, the matrix b(ei , ej ) (i,j)∈S—S is
diagonal, and b§m (eS , eS ) = aS . Therefore, as S runs over all the subsets of m
m
indices, the elements eS are anisotropic and form an orthogonal basis of V with
§m
respect to the bilinear form b .
On the other hand, if S = {1, . . . , n} S is the complement of S, we have
eS § e S = µ S e
for some µS = ±1. Since § is symmetric when m is even and skew-symmetric when
m is odd (see (??)), it follows that
µS µS = (’1)m .
(10.25)
De¬ne u on the basis elements eS by
u(eS ) = µS a’1 eS
(10.26) S
126 II. INVARIANTS OF INVOLUTIONS

m
and extend u to V by linearity. We then have
b§m u(eS ), eS e = µS e = eS § eS
and
b§m eS , u(eS ) = µS e = (’1)m eS § eS .
Moreover, if T = S, then
b u(eS ), eT = 0 = eS § eT = b eS , u(eT ) .
The equations (??) thus hold when x, y run over the basis (eS ); therefore they hold
for all x, y ∈ V by bilinearity.
n
For all S we have aS aS = i=1 b(ei , ei ), hence (??) follows from (??) and (??).
From (??), it follows that for all f ∈ EndF (V ) and all x, y ∈ V ,
b§m u(x), f (y) e = x § f (y),
hence
b§m σ §m (f ) —¦ u(x), y e = γ(f )(x) § y.
The left side also equals
b§m u —¦ u’1 —¦ σ §m (f ) —¦ u(x), y e = u’1 —¦ σ §m (f ) —¦ u (x) § y,
hence
u’1 —¦ σ §m (f ) —¦ u = γ(f ) for f ∈ EndF (V ).
Therefore, σ §m = Int(u) —¦ γ.
We next show that σ §m and γ commute. By (??), we have σ §m (u) = (’1)m u,
hence γ(u) = (’1)m u. Therefore, γ —¦ σ §m = Int(u’1 ), while σ §m —¦ γ = Int(u).
Since u2 ∈ F — , we have Int(u) = Int(u’1 ), and the claim is proved.
Finally, assume char F = 2 and m ≥ 2. The proof of (??) shows that the quad-
ratic map q may be de¬ned by partitioning the subsets of m indices in {1, . . . , n}
into two classes C, C such that the complement of every subset in C lies in C and
vice versa, and letting
q(x) = xS xS e
S∈C

xS a’1 eS , hence
for x = xS eS . By de¬nition of u, we have u(x) =
S S S

xS a’1 xS a’1 e.
q u(x) = S S
S∈C

Since aS aS = δ for all S, it follows that q u(x) = δ ’1 q(x).

10.E. De¬nition of the discriminant algebra. Let (B, „ ) be a central
simple algebra with involution of the second kind over a ¬eld F . We assume that
the degree of (B, „ ) is even: deg(B, „ ) = n = 2m. The center of B is denoted K;
it is a quadratic ´tale F -algebra with nontrivial automorphism ι. We ¬rst consider
e
the case where K is a ¬eld, postponing to the end of the section the case where
F — F . The K-algebra B is thus central simple. The K-algebra »m B has a
K
canonical involution γ, which is of the ¬rst kind, and also has the involution „ §m
induced by „ , which is of the second kind. The de¬nition of the discriminant algebra
D(B, „ ) is based on the following crucial result:
§10. THE DISCRIMINANT ALGEBRA 127


(10.27) Lemma. The involutions γ and „ §m on »m B commute. Moreover, if
char F = 2 and m ≥ 2, the canonical quadratic pair (γ, f ) on »m B satis¬es
f „ §m (x) = ι f (x)
for all x ∈ Sym(»m B, γ).
Proof : We reduce to the split case by a scalar extension. To construct a ¬eld
extension L of F such that K —F L is a ¬eld and B —F L is split, consider the
division K-algebra D which is Brauer-equivalent to B. By (??), this algebra has
an involution of the second kind θ. We may take for L a maximal sub¬eld of
Sym(D, θ).
We may thus assume B = EndK (V ) for some n-dimensional vector space V
over K and „ = σh for some nonsingular hermitian form h on V . Consider an
orthogonal basis (ei )1¤i¤n of V and let V0 ‚ V be the F -subspace of V spanned
by e1 , . . . , en . Since h(ei , ei ) ∈ F — for i = 1, . . . , n, the restriction h0 of h to
V0 is a nonsingular symmetric bilinear form which is not alternating. We have
V = V0 —F K, hence
B = EndF (V0 ) —F K.
Moreover, since „ is the adjoint involution with respect to h,
„ = „0 — ι
where „0 is the adjoint involution with respect to h0 on EndF (V0 ). Therefore, there
is a canonical isomorphism
m
»m B = EndF ( V0 ) — F K
and
„ §m = „0 — ι.
§m


On the other hand, the canonical bilinear map
m m n
§: V— V’ V
is derived by scalar extension to K from the canonical bilinear map § on m V0 ,
m
hence γ = γ0 — IdK where γ0 is the canonical involution on EndF ( V0 ). By
§m §m
Proposition (??), „0 and γ0 commute, hence „ and γ also commute.
Suppose now that char F = 2 and m ≥ 2. Let z ∈ K F . In view of the canon-
m
ical isomorphism »m B = EndF ( V0 ) —F K, every element x ∈ Sym(»m B, γ)
m
may be written in the form x = x0 — 1 + x1 — z for some x0 , x1 ∈ EndF ( V0 )
m
symmetric under γ0 . Proposition (??) yields an element u ∈ EndF ( V0 ) such
§m
that „0 = Int(u) —¦ γ0 , hence
„ §m (x) = „0 (x0 ) — 1 + „0 (x1 ) — ι(z) = (ux0 u’1 ) — 1 + (ux1 u’1 ) — ι(z).
§m §m


To prove f „ §m (x) = ι f (x) , it now su¬ces to show that f (uyu’1 ) = f (y) for
m
all y ∈ Sym EndF ( V0 ), γ0 .
m
Let q : V0 ’ F be the canonical quadratic form uniquely de¬ned (up to a
scalar multiple) by (??). Under the associated standard identi¬cation, the elements
m m m
in Sym EndF ( V0 ), γ0 correspond to symmetric tensors in V0 — V0 , and
m
we have f (v — v) = q(v) for all v ∈ V0 . Since symmetric tensors are spanned
by elements of the form v — v, it su¬ces to prove
m
f u —¦ (v — v) —¦ u’1 = f (v — v) for all v ∈ V0 .
128 II. INVARIANTS OF INVOLUTIONS


The proof of (??) shows that γ0 (u) = u and u2 = δ ’1 ∈ F — , hence
u —¦ (v — v) —¦ u’1 = δu —¦ (v — v) —¦ γ0 (u) = δu(v) — u(v);
therefore, by (??),
f u —¦ (v — v) —¦ u’1 = δq u(v) = q(v) = f (v — v),
and the proof is complete.
The lemma shows that the composite map θ = „ §m —¦ γ is an automorphism of
order 2 on the F -algebra B. Note that θ(x) = ι(x) for all x ∈ K, since „ §m is an
involution of the second kind while γ is of the ¬rst kind.
(10.28) De¬nition. The discriminant algebra D(B, „ ) of (B, „ ) is the F -subal-
gebra of θ-invariant elements in »m B. It is thus a central simple F -algebra of
degree
n
deg D(B, „ ) = deg »m B = .
m
The involutions γ and „ §m restrict to the same involution of the ¬rst kind „ on
D(B, „ ):
„ = γ|D(B,„ ) = „ §m |D(B,„ ) .
Moreover, if char F = 2 and m ≥ 2, the canonical quadratic pair (γ, f ) on »m B
restricts to a canonical quadratic pair („ , fD ) on D(B, „ ); indeed, for an element
x ∈ Sym D(B, „ ), „ we have „ §m (x) = x, hence, by (??), f (x) = ι f (x) , and
therefore f (x) ∈ F .
The following number-theoretic observation on deg D(B, „ ) is useful:
2m
(10.29) Lemma. Let m be an integer, m ≥ 1. The binomial coe¬cient m
satis¬es
2 mod 4 if m is a power of 2;
2m

m
0 mod 4 if m is not a power of 2.
Proof : For every integer m ≥ 1, let v(m) ∈ N be the exponent of the highest
power of 2 which divides m, i.e., v(m) is the 2-adic valuation of m. The equation
(m + 1) 2(m+1) = 2(2m + 1) 2m yields
m+1 m
2(m+1) 2m
v =v + 1 ’ v(m + 1) for m ≥ 1.
m+1 m

On the other hand, let (m) = m0 + · · · + mk where the 2-adic expansion of m
is m = m0 + 2m1 + 22 m2 + · · · + 2k mk with m0 , . . . , mk = 0 or 1. It is easily
seen that the function (m) satis¬es the same recurrence relation as v 2m and
m
2 2m 2m
(1) = 1 = v 1 , hence (m) = v m for all m ≥ 1. In particular, v m = 1 if m
is a power of 2, and v 2m ≥ 2 otherwise.
m

(10.30) Proposition. Multiplication in »m B yields a canonical isomorphism
D(B, „ ) —F K = »m B
such that „ — IdK = γ and „ — ι = „ §m . The index ind D(B, „ ) divides 4, and
ind D(B, „ ) = 1 or 2 if m is a power of 2.
The involution „ is of symplectic type if m is odd or char F = 2; it is of
orthogonal type if m is even and char F = 2.
§10. THE DISCRIMINANT ALGEBRA 129


Proof : The ¬rst part follows from the de¬nition of D(B, „ ) and its involution „ . By
(??) we have ind »m B = 1 or 2 since »m B and B —m are Brauer-equivalent, hence
ind D(B, „ ) divides 2[K : F ] = 4. However, if m is a power of 2, then ind D(B, „ )
cannot be 4 since the preceding lemma shows that deg D(B, „ ) ≡ 2 mod 4.
Since γ = „ — IdK , the involutions γ and „ have the same type, hence „ is
orthogonal if and only if m is even and char F = 2.

For example, if B is a quaternion algebra, i.e., n = 2, then m = 1 hence »m B =
B and „ §m = „ . The algebra D(B, „ ) is the unique quaternion F -subalgebra of B
such that B = D(B, „ ) —F K and „ = γ0 — ι where γ0 is the canonical (conjugation)
involution on D(B, „ ): see (??).
To conclude this section, we examine the case where K = F — F . We may then
assume B = E — E op for some central simple F -algebra E of degree n = 2m and
„ = µ is the exchange involution. Note that there is a canonical isomorphism

(»m E)op = EndE —m (E —m sm )op ’ End(E op )—m (E op )—m sop = »m (E op )
’ m

which identi¬es f op ∈ (»m E)op with the endomorphism of (E op )—m sop which maps
m
op
sop to (sf )op (thus, (sop )f = (sf )op ) (see Exercise ?? of Chapter I). Therefore,
m m m m
m op
the notation » E is not ambiguous. We may then set
»m B = »m E — »m E op
and de¬ne the canonical involution γ on »m B by means of the canonical involution
γE on »m E:
γ(x, y op ) = γE (x), γE (y)op for x, y ∈ »m E.
Similarly, if char F = 2 and m ≥ 2, the canonical pair (γE , fE ) on »m E (see (??))
induces a canonical quadratic pair (γ, f ) on »m B by
f (x, y op ) = fE (x), fE (y) ∈ F — F for x, y ∈ Sym(»m E, γ).
We also de¬ne the involution µ§m on »m B as the exchange involution on »m E —
»m E op :
µ§m (x, y op ) = (y, xop ) for x, y ∈ »m E.
The involutions µ§m and γ thus commute, hence their composite θ = µ§m —¦ γ is an
F -automorphism of order 2 on »m B. The invariant elements form the F -subalgebra
D(B, µ) = { x, γ(x)op | x ∈ »m E } »m E.
(10.31)
The involutions µ§m and γ coincide on this subalgebra and induce an involution
which we denote µ.
The following proposition shows that these de¬nitions are compatible with the
notions de¬ned previously in the case where K is a ¬eld:
(10.32) Proposition. Let (B, „ ) be a central simple algebra with involution of
the second kind over a ¬eld F . Suppose the center K of B is a ¬eld. The K-

algebra isomorphism • : (BK , „K ) ’ (B — B op , µ) of (??) which maps x — k to

op
xk, „ (x)k induces a K-algebra isomorphism

»m • : (»m B)K ’ »m B — »m B op

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