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i
i=1 i=1

Since q(v) = f —¦ •q (v — v), the ¬rst assertion is proved. Identifying EndF (V ) with
Mn (F ) by means of the basis (e1 , . . . , en ) maps to the matrix
« 
0
1
1 q(e2i )
¬ ·
..
=  where i = q(e
.
2i’1 ) 0
0 m

The characteristic polynomial of is the product of the characteristic polynomials
of 1 , . . . , m . This implies the second assertion.
(5.13) Example. Suppose that char F = 2 and let (1, u, v, w) be a quaternion
basis of a quaternion F -algebra Q = [a, b)F . In every quadratic pair (σ, f ) on Q,
the involution σ is symplectic. It is therefore the canonical involution γ. The space
Sym(Q, γ) is the span of 1, v, w, and Alt(Q, γ) = F . Since 1 = u + γ(u) and
TrdQ (u) = 1, the map f may be any linear form on Sym(Q, γ) such that f (1) = 1.
An element corresponding to f as in (??) is
= u + f (w)b’1 v + f (v)b’1 w.
(For a given f , the element is uniquely determined up to the addition of an element
in F .)
Quadratic pairs on tensor products. Let A1 , A2 be central simple F -
algebras. Given a quadratic pair (σ1 , f1 ) on A1 and an involution σ2 on A2 , we
aim to de¬ne a quadratic pair on the tensor product A1 —F A2 . If char F = 2, this
amounts to de¬ning an orthogonal involution on A1 —F A2 , and it su¬ces to take
σ1 — σ2 , assuming that σ2 is orthogonal, see (??). For the rest of this section, we
may thus focus on the case where char F = 2.
(5.14) Lemma. Let (A1 , σ1 ) and (A2 , σ2 ) be central simple algebras with involu-
tion of the ¬rst kind over a ¬eld F of characteristic 2.
(5.15) Symd(A1 , σ1 ) — Symd(A2 , σ2 ) =
Symd(A1 , σ1 ) — Sym(A2 , σ2 ) © Sym(A1 , σ1 ) — Symd(A2 , σ2 );

(5.16) Symd(A1 — A2 , σ1 — σ2 ) © Sym(A1 , σ1 ) — Sym(A2 , σ2 ) =
Symd(A1 , σ1 ) — Sym(A2 , σ2 ) + Sym(A1 , σ1 ) — Symd(A2 , σ2 );
§5. QUADRATIC FORMS 61


(5.17) Sym(A1 — A2 , σ1 — σ2 ) =
Symd(A1 — A2 , σ1 — σ2 ) + Sym(A1 , σ1 ) — Sym(A2 , σ2 ) .
Proof : Equation (??) is clear. For x1 ∈ A1 and s2 ∈ Sym(A2 , σ2 ),
x1 + σ1 (x1 ) — s2 = x1 — s2 + (σ1 — σ2 )(x1 — s2 ),
hence Symd(A1 , σ1 ) — Sym(A2 , σ2 ) ‚ Symd(A1 — A2 , σ1 — σ2 ). Similarly,
Sym(A1 , σ1 ) — Symd(A2 , σ2 ) ‚ Symd(A1 — A2 , σ1 — σ2 ),
hence the left side of (??) contains the right side. To prove the reverse inclusion,
consider x ∈ A1 — A2 . If x + (σ1 — σ2 )(x) ∈ Sym(A1 , σ1 ) — Sym(A2 , σ2 ), then
x + (σ1 — σ2 )(x) is invariant under σ1 — IdA2 , hence
x + (σ1 — IdA2 )(x) + (IdA1 — σ2 )(x) + (σ1 — σ2 )(x) = 0.
Therefore, the element u = x + (σ1 — IdA2 )(x) is invariant under IdA1 — σ2 , hence it
lies in A1 — Sym(A2 , σ2 ). Similarly, the element v = (σ1 — IdA2 )(x) + (σ1 — σ2 )(x)
is in Sym(A1 , σ1 ) — A2 . On the other hand, it is clear by de¬nition that u ∈
Symd(A1 , σ1 ) — A2 and v ∈ A1 — Symd(A2 , σ2 ), hence
u ∈ Symd(A1 , σ1 ) — Sym(A2 , σ2 ) and v ∈ Sym(A1 , σ1 ) — Symd(A2 , σ2 ).
Since x + (σ1 — σ2 )(x) = u + v, the proof of (??) is complete.
Since the left side of equation (??) obviously contains the right side, it suf-
¬ces to prove that both sides have the same dimension. Let ni = deg Ai , so that
dimF Sym(Ai , σi ) = 2 ni (ni + 1) and dimF Symd(Ai , σi ) = 1 ni (ni ’ 1) for i = 1, 2.
1
2
From (??), it follows that
dimF Symd(A1 , σ1 ) — Sym(A2 , σ2 ) + Sym(A1 , σ1 ) — Symd(A2 , σ2 ) =
1
’ 1)(n2 + 1) + 1 n1 n2 (n1 + 1)(n2 ’ 1) ’ 1 n1 n2 (n1 ’ 1)(n2 ’ 1)
4 n1 n2 (n1 4 4
= 1 n1 n2 (n1 n2 + n1 + n2 ’ 3).
4
Therefore, (??) yields
dimF Symd(A1 — A2 , σ1 — σ2 ) + Sym(A1 , σ1 ) — Sym(A2 , σ2 ) =
1
’ 1) + 1 n1 n2 (n1 + 1)(n2 + 1) ’ 1 n1 n2 (n1 n2 + n1 + n2 ’ 3)
2 n1 n2 (n1 n2 4 4
= 1 n1 n2 (n1 n2 + 1) = dimF Sym(A1 — A2 , σ1 — σ2 ).
2



(5.18) Proposition. Suppose that char F = 2. Let (σ1 , f1 ) be a quadratic pair
on a central simple F -algebra A1 and let (A2 , σ2 ) be a central simple F -algebra
with involution of the ¬rst kind. There is a unique quadratic pair (σ1 — σ2 , f1— ) on
A1 —F A2 such that
f1— (s1 — s2 ) = f1 (s1 ) TrdA2 (s2 )
for s1 ∈ Sym(A1 , σ1 ) and s2 ∈ Sym(A2 , σ2 ).
Proof : Since σ1 is symplectic, (??) shows that σ1 — σ2 is symplectic. To prove the
existence of a quadratic pair (σ1 —σ2 , f1— ) as above, we have to show that the values
that f1— is required to take on Symd(A1 — A2 , σ1 — σ2 ) because of the quadratic
pair conditions agree with the prescribed values on Sym(A1 , σ1 ) — Sym(A2 , σ2 ). In
view of the description of Symd(A1 — A2 , σ1 — σ2 ) © Sym(A1 , σ1 ) — Sym(A2 , σ2 )
in the preceding lemma, it su¬ces to consider the values of f1— on elements of the
62 I. INVOLUTIONS AND HERMITIAN FORMS


form x1 + σ1 (x1 ) — s2 = x1 — s2 + (σ1 — σ2 )(x1 — s2 ) or s1 — x2 + σ2 (x2 ) =
s1 — x2 + (σ1 — σ2 )(s1 — x2 ) with xi ∈ Ai and si ∈ Sym(Ai , σi ) for i = 1, 2. Since
(σ1 , f1 ) is a quadratic pair on A1 , we have
f1 x1 + σ1 (x1 ) TrdA2 (s2 ) = TrdA1 (x1 ) TrdA2 (s2 ) = TrdA1 —A2 (x1 — s2 ),
as required. For the second type of element we have
f1 (s1 ) TrdA2 x2 + σ2 (x2 ) = 0.
On the other hand, since σ1 is symplectic we have TrdA1 (s1 ) = 0, hence
TrdA1 —A2 (s1 — x2 ) = TrdA1 (s1 ) TrdA2 (x2 ) = 0.
Therefore,
f1 (s1 ) TrdA2 x2 + σ2 (x2 ) = TrdA1 —A2 (s1 — x2 )
for s1 ∈ Sym(A1 , σ1 ) and x2 ∈ A2 , and the existence of the quadratic pair (σ1 —
σ2 , f1— ) is proved.
Uniqueness is clear, since the values of the linear map f1— are determined on
the set Symd(A1 — A2 , σ1 — σ2 ) and on Sym(A1 , σ1 ) — Sym(A2 , σ2 ), and (??) shows
that these subspaces span Sym(A1 — A2 , σ1 — σ2 ).

(5.19) Example. Let (V1 , q1 ) be a nonsingular quadratic space of even dimen-
sion and let (V2 , b2 ) be a nonsingular symmetric bilinear space over a ¬eld F of
characteristic 2. Let (σ1 , f1 ) be the quadratic pair on A1 = EndF (V1 ) associ-
ated with q1 (see (??)) and let σ2 = σb2 denote the adjoint involution with re-
spect to b2 on A2 = EndF (V2 ). We claim that, under the canonical isomorphism
A1 — A2 = EndF (V1 — V2 ), the quadratic pair (σ1 — σ2 , f1— ) is associated with the
quadratic form q1 — b2 on V1 — V2 whose polar form is bq1 — b2 and such that
(q1 — b2 )(v1 — v2 ) = q1 (v1 )b2 (v2 , v2 ) for v1 ∈ V1 and v2 ∈ V2 .

Indeed, letting •1 , •2 and • denote the standard identi¬cations V1 — V1 ’ ’
∼ ∼
EndF (V1 ), V2 — V2 ’ EndF (V2 ) and (V1 — V2 ) — (V1 — V2 ) ’ EndF (V1 — V2 )
’ ’
associated with the bilinear forms bq1 , b2 and bq1 — b2 , we have
•(v1 — v2 — v1 — v2 ) = •1 (v1 — v1 ) — •2 (v2 — v2 )

and

f1 •1 (v1 — v1 ) TrdA2 •2 (v2 — v2 ) = q1 (v1 )b2 (v2 , v2 ),

hence

f1— •(v1 — v2 — v1 — v2 ) = q1 — b2 (v1 — v2 ).
(5.20) Corollary. Let (A1 , σ1 ), (A2 , σ2 ) be central simple algebras with symplectic
involutions over a ¬eld F of arbitrary characteristic. There is a unique quadratic
pair (σ1 — σ2 , f— ) on A1 — A2 such that f— (s1 — s2 ) = 0 for all s1 ∈ Skew(A1 , σ1 ),
s2 ∈ Skew(A2 , σ2 ).
1
Proof : If char F = 2, the linear form f— which is the restriction of TrdA1 —A2 to
2
Sym(A1 — A2 , σ1 — σ2 ) satis¬es
1
f— (s1 — s2 ) = TrdA1 (s1 ) TrdA2 (s2 ) = 0
2
EXERCISES 63


for all s1 ∈ Skew(A1 , σ1 ), s2 ∈ Skew(A2 , σ2 ). Suppose next that char F = 2.
For any linear form f1 on Sym(A1 , σ1 ) we have f1 (s1 ) TrdA2 (s2 ) = 0 for all s1 ∈
Sym(A1 , σ1 ), s2 ∈ Sym(A2 , σ2 ), since σ2 is symplectic. Therefore, we may set
(σ1 — σ2 , f— ) = (σ1 — σ2 , f1— )
for any quadratic pair (σ1 , f1 ) on A1 . Uniqueness of f— follows from (??).

(5.21) De¬nition. The quadratic pair (σ1 — σ2 , f— ) of (??) is called the canonical
quadratic pair on A1 — A2 .




Exercises
1. Let A be a central simple algebra over a ¬eld F and ¬x a ∈ A. Show that there
is a canonical F -algebra isomorphism EndA (aA) EndA (Aa) which takes f ∈
EndA (aA) to the endomorphism f ∈ EndA (Aa) de¬ned by (xa)f = xf (a) for
x ∈ A, and the inverse takes g ∈ EndA (Aa) to the endomorphism g ∈ EndA (aA)
de¬ned by g(ax) = ag x for x ∈ A.

Show that there is a canonical F -algebra isomorphism EndA (Aa)op ’ ’
EndAop (aop Aop ) which, for f ∈ EndA (Aa), maps f op to the endomorphism
˜ ˜
f de¬ned by f (mop ) = (mf )op . Therefore, there is a canonical isomorphism
op
EndAop (Aop aop ). Use it to identify (»k A)op = »k (Aop ), for
EndA (Aa)
k = 1, . . . , deg A.
2. Let Q be a quaternion algebra over a ¬eld F of arbitrary characteristic. Show
that the conjugation involution is the only linear map σ : Q ’ Q such that
σ(1) = 1 and σ(x)x ∈ F for all x ∈ F .
3. (Rowen-Saltman [?]) Let V be a vector space of dimension n over a ¬eld F and
let „ be an involution of the ¬rst kind on EndF (V ). Prove that „ is orthogonal
if and only if there exist n symmetric orthogonal10 idempotents in EndF (V ).
Find a similar characterization of the symplectic involutions on EndF (V ).
4. Let A be a central simple algebra with involution σ of the ¬rst kind. Show
that σ is orthogonal if and only if it restricts to the identity on a maximal ´tale
e
(commutative) subalgebra of A.
Hint: Extend scalars and use the preceding exercise.
5. Show that in a central simple algebra with involution, every left or right ideal is
generated by a symmetric element, unless the algebra is split and the involution
is symplectic.
6. (Albert) Let b be a symmetric, nonalternating bilinear form on a vector space V
over a ¬eld of characteristic 2. Show that V contains an orthogonal basis for b.
7. Let (ai )i=1,...,n2 be an arbitrary basis of a central simple algebra A, and let
(bi )i=1,...,n2 be the dual basis for the bilinear form TA , which means that
TrdA (ai bj ) = δij for i, j = 1, . . . , n2 . Show that the Goldman element of
n2
A is ai — b i .
i=1
Hint: Reduce by scalar extension to the split case and show that it su¬ces
to prove the assertion for the standard basis of Mn (F ).

10 Two idempotents e, f are called orthogonal if ef = f e = 0.
64 I. INVOLUTIONS AND HERMITIAN FORMS


8. Let (1, i, j, k) be a quaternion basis in a quaternion algebra Q of characteristic
di¬erent from 2. Show that the Goldman element in Q — Q is g = 1 (1 — 1 + i —
2
i’1 + j — j ’1 + k — k ’1 ). Let (1, u, v, w) be a quaternion basis in a quaternion
algebra Q of characteristic 2. Show that the Goldman element in Q — Q is
g = 1 — 1 + u — 1 + 1 — u + w — v ’1 + v ’1 — w.
9. Let K/F be a quadratic extension of ¬elds of characteristic di¬erent from 2,
and let a ∈ F — , b ∈ K — . Prove the “projection formula” for the norm of the
quaternion algebra (a, b)K :
NK/F (a, b)K ∼ a, NK/F (b) F
(where ∼ denotes Brauer-equivalence). Prove corresponding statements in
characteristic 2: if K/F is a separable quadratic extension of ¬elds and a ∈ F ,
b ∈ K — , c ∈ K, d ∈ F — ,
NK/F [a, b)K ∼ a, NK/F (b) and NK/F [c, d)K ∼ TK/F (c), d .
F F
Hint (when char F = 2): Let ι be the non-trivial automorphism of K/F
and let (1, i1 , j1 , k1 ) (resp. (1, i2 , j2 , k2 )) denote the usual quaternion basis of
a, ι(b) K = ι (a, b)K (resp. (a, b)K ). Let s be the switch map on ι (a, b)K —K
1
(a, b)K . Let u = 2 1 + a’1i1 — i2 + b’1 j1 — j2 ’ (ab)’1 k1 — k2 ∈ a, ι(b) K —K
(a, b)K . Show that s(u)u = 1, and that s = Int(u) —¦ s is a semi-linear auto-
morphism of order 2 of ι (a, b)K —K (a, b)K which leaves invariant i1 — 1, 1 — i2
and j1 — j2 . Conclude that the F -subalgebra of elements invariant under s is
Brauer-equivalent to a, NK/F (b) F . If b ∈ F , let v = 1 + u; if b ∈ F , pick
c ∈ K such that c2 ∈ F and let v = c + uι(c). Show that u = s (v)’1 v and
that Int(v) maps the subalgebra of s-invariant elements onto the subalgebra of
s -invariant elements.
10. (Knus-Parimala-Srinivas [?, Theorem 4.1]) Let A be a central simple algebra
over a ¬eld F , let V be an F -vector space and let
ρ : A —F A ’ EndF (V )
be an isomorphism of F -algebras. We consider V as a left A-module via av =
ρ(a — 1)(v) and identify EndA (A) = A, HomA (A, V ) = V by mapping every
homomorphism f to 1f . Moreover, we identify EndA (V ) = Aop by setting
op
va = ρ(1 — a)(v) for v ∈ V , a ∈ A
and HomA (V, A) = V — = HomF (V, F ) by mapping h ∈ HomA (V, A) to the
linear form
v ’ TrdA (v h ) for v ∈ V .
Let
EndA (A) HomA (A, V ) A V
B = EndA (A • V ) = = ;
V— Aop
HomA (V, A) EndA (V )
this is a central simple F -algebra which is Brauer-equivalent to A, by Proposi-
tion (??). Let γ = ρ(g) ∈ EndF (V ), where g ∈ A—F A is the Goldman element.
Show that
av b γ(v)

op
γ (f ) aop
t
fb
is an involution of orthogonal type of B.
EXERCISES 65


11. (Knus-Parimala-Srinivas [?, Theorem 4.2]) Let K/F be a separable quadratic
extension with nontrivial automorphism ι, let A be a central simple K-algebra,
let V be an F -vector space and let
ρ : NK/F (A) ’ EndF (V )
be an isomorphism of F -algebras. Set W = V —F K and write
ρ : A —K ιA ’ EndK (W )
for the isomorphism induced from ρ by extension of scalars. We consider W as
a left A-module via av = ρ(a — 1)(v). Let
EndA (A) HomA (A, W ) A W
B = EndA (A • W ) = =
W— ι op
HomA (W, A) EndA (W ) A
with identi¬cations similar to those in Exercise ??. The K-algebra B is Brauer
equivalent to A by (??). Show that
a w b ι(w)

ι op ι op
f b ι(f ) a
is an involution of the second kind of B.
12. Let A be a central simple F -algebra with involution σ of the ¬rst kind. Recall
the F -algebra isomorphism
σ— : A —F A ’ EndF A
de¬ned in the proof of Corollary (??) by
σ— (a — b)(x) = axσ(b) for a, b, x ∈ A.
Show that the image of the Goldman element g ∈ A—A under this isomorphism
is δσ where δ = +1 if σ is orthogonal and δ = ’1 if σ is symplectic. Use this
result to de¬ne canonical F -algebra isomorphisms
op
EndF A/ Alt(A, σ) EndF Sym(A, σ) if δ = +1,
2
sA op
EndF A/ Symd(A, σ) EndF Skew(A, σ) if δ = ’1,
EndF Alt(A, σ) if δ = +1,
»2 A
EndF Symd(A, σ) if δ = ’1.
13. (Saltman [?, Proposition 5]) Let A, B be central simple algebras of degrees m, n
over a ¬eld F . For every F -algebra homomorphism f : A ’ B, de¬ne a map
f : Aop —F B ’ B by f (aop — b) = f (a)b. Show that f is a homomorphism of
right Aop —F B-modules if B is endowed with the following Aop —F B-module
structure:
x —f (aop — b) = f (a)xb for a ∈ A and b, x ∈ B.
Show that ker f ‚ Aop —F B is a right ideal of reduced dimension mn ’ (n/m)
generated by the elements aop — 1 ’ 1 — f (a) for a ∈ A and that
Aop —F B = (1 — B) • ker f .
Conversely, show that every right ideal I ‚ Aop —F B of reduced dimension
mn ’ (n/m) such that
Aop —F B = (1 — B) • I
de¬nes an F -algebra homomorphism f : A ’ B such that I = ker f .
66 I. INVOLUTIONS AND HERMITIAN FORMS


Deduce from the results above that there is a natural one-to-one correspon-
dence between the set of F -algebra homomorphisms A ’ B and the rational
points in an open subset of the Severi-Brauer variety SBd (Aop —F B) where
d = mn ’ (n/m).
14. Let (A, σ) be a central simple algebra with involution of the ¬rst kind over
a ¬eld F of characteristic di¬erent from 2. Let a ∈ A be an element whose
minimal polynomial over F is separable. Show that a is symmetric for some
symplectic involution on A if and only if its reduced characteristic polynomial
is a square.
15. Let (A, σ) be a central simple algebra with involution of orthogonal type. Show
that every element in A is the product of two symmetric elements, one of which
is invertible.
Hint: Use (??).
16. Let V be a ¬nite dimensional vector space over a ¬eld F of arbitrary character-
istic and let a ∈ EndF (V ). Extend the notion of involution trace by using the
structure of V as an F [a]-module to de¬ne a nonsingular symmetric bilinear
form b : V — V ’ F such that a is invariant under the adjoint involution σb .
17. Let K/F be a separable quadratic extension of ¬elds with nontrivial automor-
phism ι. Let V be a vector space of dimension n over K and let b ∈ EndK (V )
be an endomorphism whose minimal polynomial has degree n and coe¬cients
in F . Show that V is a free F [b]-module of rank 1 and de¬ne a nonsingular her-
mitian form h : V —V ’ K such that b is invariant under the adjoint involution
σh .
Show that a matrix m ∈ Mn (K) is symmetric under some involution of the
second kind whose restriction to K is ι if and only if all the invariant factors
of m have coe¬cients in F .
18. Let q be a nonsingular quadratic form of dimension n over a ¬eld F , with n even
if char F = 2, and let a ∈ Mn (F ) be a matrix representing q, in the sense that
q(X) = X · a · X t . After identifying Mn (F ) with EndF (F n ) by mapping every
matrix m ∈ Mn (F ) to the endomorphism x ’ m · x, show that the quadratic
pair (σq , fq ) on EndF (F n ) associated to the quadratic map q : F n ’ F is the
same as the quadratic pair (σa , fa ) associated to a.
19. Let Q1 , Q2 be quaternion algebras with canonical involutions γ1 , γ2 over a ¬eld
F of arbitrary characteristic. Show that Alt(Q1 —Q2 , γ1 —γ2 ) = { q1 —1’1—q2 |
TrdQ1 (q1 ) = TrdQ2 (q2 ) }. If char F = 2, show that f (q1 — 1 + 1 — q2 ) =
TrdQ1 (q1 ) = TrdQ2 (q2 ) for all q1 — 1 + 1 — q2 ∈ Symd(Q1 — Q2 , γ1 — γ2 ) and for
all quadratic pairs (γ1 — γ2 , f ) on Q1 — Q2 .
20. The aim of this exercise is to give a description of the variety of quadratic
pairs on a central simple algebra in the spirit of (??). Let σ be a symplectic
involution on a central simple algebra A over a ¬eld F of characteristic 2 and
let σ— : A — A ’ EndF (A) be the isomorphism of Exercise ??. Let Iσ ‚ A — A
denote the right ideal corresponding to σ by (??) and let J ‚ A — A be the left
ideal generated by 1 ’ g, where g is the Goldman element. Denote by A0 ‚ A
the kernel of the reduced trace: A0 = { a ∈ A | TrdA (a) = 0 }. Show that
σ— (Iσ ) = Hom(A, A0 ), σ— (J ) = Hom A/ Sym(A, σ), A
and
σ— (J 0 ) = Hom A, Sym(A, σ) .
NOTES 67


Let now I ‚ A — A be a left ideal containing J , so that
σ— (I) = Hom(A/U, A)
for some vector space U ‚ Sym(A, σ). Show that σ— I·(1+g) = Hom(A/W, A),
where W = { a ∈ A | a + σ(a) ∈ U }, and deduce that σ— [I · (1 + g)]0 = Iσ if
and only if W = A0 , if and only if U © Symd(A, σ) = ker Trpσ .
Observe now that the set of rational points in SB s2 (Aop ) is in canonical
one-to-one correspondence with the set of left ideals I ‚ A — A containing J 0
and such that rdim I ’ rdim J = 1. Consider the subset U of such ideals which
satisfy [I · (1 + g)]0 • (1 — A) = A — A. Show that the map which carries every
’1
quadratic pair (σ, f ) on A to the left ideal σ— Hom(A/ ker f, A) de¬nes a
bijection from the set of quadratic pairs on A onto U.
Hint: For I ∈ U, the right ideal [I · (1 + g)]0 corresponds by (??) to
some symplectic involution σ. If U ‚ A is the subspace such that σ— (I) =
Hom(A/U, A), there is a unique quadratic pair (σ, f ) such that U = ker f .
21. Let (V1 , b1 ) and (V2 , b2 ) be vector spaces with nonsingular alternating forms
over an arbitrary ¬eld F . Show that there is a unique quadratic form q on
V1 — V2 whose polar form is b1 — b2 and such that q(v1 — v2 ) = 0 for all
v1 ∈ V1 , v2 ∈ V2 . Show that the canonical quadratic pair (σb1 — σb2 , f— ) on
EndF (V1 ) — EndF (V2 ) = EndF (V1 — V2 ) is associated with the quadratic form
q.
22. Let (A, σ) be a central simple algebra with involution of the ¬rst kind over an
arbitrary ¬eld F . Assume σ is symplectic if char F = 2. By (??), there exists
an element ∈ A such that +σ( ) = 1. De¬ne a quadratic form q(A,σ) : A ’ F
by
q(A,σ) (x) = TrdA σ(x) x for x ∈ A.
Show that this quadratic form does not depend on the choice of such that
+ σ( ) = 1. Show that the associated quadratic pair on EndF (A) corresponds
to the canonical quadratic pair (σ — σ, f— ) on A — A under the isomorphism

σ— : A — A ’ EndF (A) such that σ— (a — b)(x) = axσ(b) for a, b, x ∈ A.





Notes
§??. Additional references for the material in this section include the classical
books of Albert [?], Deuring [?] and Reiner [?]. For Severi-Brauer varieties, see
Artin™s notes [?]. A self-contained exposition of Severi-Brauer varieties can be
found in Jacobson™s book [?, Chapter 3].
§??. The ¬rst systematic investigations of involutions of central simple alge-
bras are due to Albert. His motivation came from the theory of Riemann matrices:
on a Riemann surface of genus g, choose a basis (ω± )1¤±¤g of the space of holo-
morphic di¬erentials and a system of closed curves (γβ )1¤β¤2g which form a basis
of the ¬rst homology group, and consider the matrix of periods P = ( γβ ω± ). This
is a complex g — 2g matrix which satis¬es Riemann™s period relations: there exists

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