1

dimF I = dimF A, σ(I)I = {0}, f I © Sym(A, σ) = {0}.

2

If char F = 2, the map f is determined by σ and ind(A, σ, f ) = ind(A, σ), hence

(σ, f ) is hyperbolic if and only if σ is hyperbolic. If char F = 2, the involution

σ is symplectic and ind(A, σ, f ) ‚ ind(A, σ); therefore, σ is hyperbolic if (σ, f ) is

hyperbolic.

We proceed to show that the quadratic pair associated to a quadratic space is

hyperbolic if and only if the quadratic space is hyperbolic.

Recall that a quadratic space over a ¬eld F is called hyperbolic if it is isometric

to a space H(U ) = (U — • U, qU ) for some vector space U where U — = HomF (U, F )

and

qU (• + u) = •(u)

for • ∈ U — and u ∈ U . The corresponding symmetric bilinear space is thus the

hyperbolic space denoted H1 (U ) above.

(6.13) Proposition. Let (V, q) be a nonsingular quadratic space of even dimension

over an arbitrary ¬eld F . The corresponding quadratic pair (σ q , fq ) on EndF (V ) is

hyperbolic if and only if the space (V, q) is hyperbolic.

Proof : By (??), the quadratic pair (σq , fq ) is hyperbolic if and only if V contains

a totally isotropic subspace U of dimension dim U = 1 dim V . This condition is

2

equivalent to (V, q) H(U ): see Scharlau [?, p. 12] (if char F = 2) and [?, p. 340]

(if char F = 2).

Hyperbolic quadratic pairs can also be characterized by the existence of certain

idempotents:

78 II. INVARIANTS OF INVOLUTIONS

(6.14) Proposition. A quadratic pair (σ, f ) on a central simple algebra A of even

degree over a ¬eld F of arbitrary characteristic is hyperbolic if and only if A contains

an idempotent e such that

f (s) = TrdA (es) for all s ∈ Sym(A, σ).

Proof : If f (s) = TrdA (es) for all s ∈ Sym(A, σ), then σ(e) = 1 ’ e, by (??),

hence the ideal eA is isotropic for σ and has reduced dimension 1 deg A if e is

2

an idempotent. Moreover, for s ∈ eA © Sym(A, σ) there exists x ∈ A such that

s = ex = σ(x)σ(e), hence

f (s) = TrdA eσ(x)σ(e) = TrdA σ(e)eσ(x) = 0.

Therefore the ideal eA is isotropic for (σ, f ), and (σ, f ) is hyperbolic.

Conversely, suppose that I ‚ A is an isotropic right ideal of reduced dimension

1

2 deg A. By arguing as in (??), we get an idempotent e0 ∈ A such that σ(e0 ) = 1’e0

1

and I = e0 A. If char F = 2, we have e0 ’ 2 ∈ Skew(A, σ) = Alt(A, σ), hence,

1

by (??), TrdA (e0 s) = 2 TrdA (s) = f (s) for all s ∈ Sym(A, σ). We may thus set

e = e0 if char F = 2, and assume that char F = 2 for the rest of the proof.

For all x ∈ Skew(A, σ), the element e = e0 ’ e0 xσ(e0 ) also is an idempotent

such that σ(e) = 1 ’ e. To complete the proof in the case where char F = 2, we

show that e satis¬es the required condition for a suitable choice of x. Consider the

linear form • on Sym(A, σ) de¬ned by

•(s) = f (s) ’ TrdA (e0 s) for s ∈ Sym(A, σ).

This form vanishes on Symd(A, σ), since σ(e0 ) = 1 ’ e0 , and also on I © Sym(A, σ)

since for all s ∈ I © Sym(A, σ) we have s2 = 0, hence TrdA (s) = 0. On the other

hand, for all x ∈ e0 Skew(A, σ)σ(e0 ), the linear form ψx ∈ Sym(A, σ)— de¬ned

by ψx (s) = TrdA (xs) also vanishes on Symd(A, σ), because x ∈ Skew(A, σ), and

on I © Sym(A, σ) because σ(e0 )e0 = 0. If we show that • = ψx for some x ∈

e0 Skew(A, σ)σ(e0 ), then we may set e = e0 + x.

We may thus complete the proof by dimension count: if deg A = n = 2m, it

can be veri¬ed by extending scalars to a splitting ¬eld of A that

dim Symd(A, σ) + I © Sym(A, σ) = mn,

hence the dimension of the space of linear forms on Sym(A, σ) which vanish on

Symd(A, σ) and I © Sym(A, σ) is m. On the other hand, by (??), the kernel of

the map which carries x ∈ e0 Skew(A, σ)σ(e0 ) to ψx ∈ Sym(A, σ)— is the space

Alt(A, σ) © e0 Skew(A, σ)σ(e0 ), and we may compute its dimension over a splitting

¬eld:

1

dim e0 Skew(A, σ)σ(e0 ) = 2 m(m + 1)

and

dim Alt(A, σ) © e0 Skew(A, σ)σ(e0 ) = 1 m(m ’ 1).

2

Therefore, the space of linear forms on Sym(A, σ) which vanish on the intersection

Symd(A, σ) + I © Sym(A, σ) is { ψx | x ∈ e0 Skew(A, σ)σ(e0 ) }, hence • = ψx for

a suitable element x ∈ e0 Skew(A, σ)σ(e0 ).

§6. THE INDEX 79

6.C. Odd-degree extensions. Using the fact that the torsion in the Witt

group of central simple algebras with involution is 2-primary (Scharlau [?]), we show

in this section that involutions which are not hyperbolic do not become hyperbolic

when tensored with a central simple algebra with involution of odd degree, nor after

an odd-degree scalar extension. This last statement generalizes a weak version of

a theorem of Springer; it is due to Bayer-Fluckiger-Lenstra [?].

Since some of the Witt group arguments do not hold in characteristic 2, we

assume that the characteristic of the base ¬eld F is di¬erent from 2 throughout

this subsection.

(6.15) Proposition. Let (A, σ) be a central simple algebra with involution (of any

kind ) over a ¬eld F and let (B, „ ) be a central simple algebra of odd degree with

involution of the ¬rst kind over F . If (A, σ) —F (B, „ ) is hyperbolic, then (A, σ) is

hyperbolic.

Proof : It follows from (??) that the algebra B is split. Let B = EndF (W ) for

some odd-dimensional F -vector space W and let b be a nonsingular symmetric

bilinear form on W such that „ = σb . Similarly, let (A, σ) = EndD (V ), σh for

some hermitian or skew-hermitian space (V, h) over a central division algebra with

involution (D, θ). We then have

(A, σ) —F (B, „ ) = EndD (V — W ), σh—b ,

and it remains to show that (V, h) is hyperbolic if (V — W, h — b) is hyperbolic.

We mimic the proof of Corollary 2.6.5 in Scharlau [?]. Suppose there exists a

non-hyperbolic hermitian or skew-hermitian space (V, h) over (D, θ) which becomes

hyperbolic when tensored by a nonsingular symmetric bilinear space (W, b) of odd

dimension. Among all such examples, choose one where the dimension of W is

minimal. Let dim W = n (≥ 3). We may assume that b has a diagonalization

1, a2 , . . . , an . Since h — b is hyperbolic, we have in the Witt group W » (D, θ)

where » = +1 if h is hermitian and » = ’1 if h is skew-hermitian,

h — a3 , . . . , an = h — ’1, ’a2 .

Since 1, ’a2 — ’1, ’a2 is hyperbolic, it follows that 1, ’a2 — h — a3 , . . . , an

is hyperbolic. By minimality of n, it follows that 1, ’a2 — h is hyperbolic, hence

h h — a2 .

Similarly, we have h h — ai for all i = 2, . . . , n, hence

n · h = h — 1, a2 , . . . , an .

By hypothesis, this form is hyperbolic; therefore, h has odd order in the Witt group

W » (D, θ), contrary to Scharlau™s result [?, Theorem 5.1].

(6.16) Corollary. Let (A, σ) be a central simple algebra with involution (of any

kind ) over a ¬eld F of characteristic di¬erent from 2 and let L/F be a ¬eld extension

of odd degree. If (AL , σL ) is hyperbolic, then (A, σ) is hyperbolic.

Proof : Embed L ’ EndF (L) by mapping x ∈ L to multiplication by x and let ν

be an involution on EndF (L) leaving the image of L elementwise invariant. (The

existence of such an involution ν follows from (??); explicitly, one may pick any

nonzero F -linear map : L ’ F and take for ν the adjoint involution with respect

to the bilinear form b(x, y) = (xy) on L.) If (AL , σL ) is hyperbolic, then (??) shows

that (A, σ) —F EndF (L), ν is hyperbolic, hence (A, σ) is hyperbolic by (??).

80 II. INVARIANTS OF INVOLUTIONS

(6.17) Corollary. Let (A, σ) be a central simple algebra with involution (of any

kind ) over a ¬eld F of characteristic di¬erent from 2 and let L/F be a ¬eld extension

of odd degree. Let u ∈ Sym(A, σ) © A— be a symmetric unit. If there exists v ∈ AL

such that u = vσL (v), then there exists w ∈ A such that u = wσ(w).

Proof : Consider the involution νu on M2 (A) as in (??). The preceding corollary

shows that M2 (A), νu is hyperbolic if M2 (A)L , (νu )L is hyperbolic. Therefore,

the corollary follows from (??).

This result has an equivalent formulation in terms of hermitian forms, which is

the way it was originally stated by Bayer-Fluckiger and Lenstra [?, Corollary 1.14]:

(6.18) Corollary (Bayer-Fluckiger-Lenstra). Let h, h be nonsingular hermitian

forms on a vector space V over a division F -algebra D, where char F = 2. The

forms h, h are isometric if they are isometric after an odd-degree scalar extension

of F .

Proof : The forms h and h are isometric if and only if h ⊥ ’h is hyperbolic, so

that the assertion follows from (??).

§7. The Discriminant

The notion of discriminant considered in this section concerns involutions of

orthogonal type and quadratic pairs. The idea is to associate to every orthogonal

involution σ over a central simple F -algebra a square class disc σ ∈ F — /F —2 , in

such a way that for the adjoint involution σb with respect to a symmetric bilinear

form b, the discriminant disc σb is the discriminant of the form b. If char F = 2,

we also associate to every quadratic pair (σ, f ) an element disc(σ, f ) ∈ F/„˜(F ),

generalizing the discriminant (Arf invariant) of quadratic forms.

7.A. The discriminant of orthogonal involutions. Let F be a ¬eld of

arbitrary characteristic. Recall that if b is a nonsingular bilinear form on a vector

space V over F , the determinant of b is the square class of the determinant of the

Gram matrix of b with respect to an arbitrary basis (e1 , . . . , en ) of V :

· F —2 ∈ F — /F —2 .

det b = det b(ei , ej ) 1¤i,j¤n

The discriminant of b is the signed determinant:

disc b = (’1)n(n’1)/2 det b ∈ F — /F —2

where n = dim V .

If dim V is odd, then for ± ∈ F — we have disc(±b) = ± disc b. Therefore, the

discriminant is an invariant of the equivalence class of b modulo scalar factors if

and only if the dimension is even. Since involutions correspond to such equivalence

classes, the discriminant of an orthogonal involution is de¬ned only for central

simple algebras of even degree.

The de¬nition of the discriminant of an orthogonal involution is based on the

following crucial result:

(7.1) Proposition. Let (A, σ) be a central simple algebra with orthogonal involu-

tion over F . If deg A is even, then for any a, b ∈ Alt(A, σ) © A— ,

NrdA (a) ≡ NrdA (b) mod F —2 .

§7. THE DISCRIMINANT 81

Proof : Fix some a, b ∈ Alt(A, σ) © A— . The involution σ = Int(a) —¦ σ is symplectic

by (??). The same proposition shows that ab ∈ Alt(A, σ ) if char F = 2 and ab ∈

Sym(A, σ ) if char F = 2; therefore, it follows from (??) that NrdA (ab) ∈ F —2 .

An alternate proof is given in (??) below.

This proposition makes it possible to give the following de¬nition:

(7.2) De¬nition. Let σ be an orthogonal involution on a central simple algebra A

of even degree n = 2m over a ¬eld F . The determinant of σ is the square class of

the reduced norm of any alternating unit:

det σ = NrdA (a) · F —2 ∈ F — /F —2 for a ∈ Alt(A, σ) © A—

and the discriminant of σ is the signed determinant:

disc σ = (’1)m det σ ∈ F — /F —2 .

The following properties follow from the de¬nition:

(7.3) Proposition. Let A be a central simple algebra of even degree over a ¬eld F

of arbitrary characteristic.

(1) Suppose σ is an orthogonal involution on A, and let u ∈ A— . If Int(u) —¦ σ is

an orthogonal involution on A, then disc Int(u) —¦ σ = NrdA (u) · disc σ.

(2) Suppose σ is a symplectic involution on A, and let u ∈ A— . If Int(u) —¦ σ is an

orthogonal involution on A, then disc Int(u) —¦ σ = NrdA (u).

(3) If A = EndF (V ) and σb is the adjoint involution with respect to some nonsin-

gular symmetric bilinear form b on V , then disc σb = disc b.

(4) Suppose σ is an orthogonal involution on A. If (B, „ ) is a central simple F -

algebra with orthogonal involution, then

disc σ if deg B is odd,

disc(σ — „ ) =

1 if deg B is even.

(5) Suppose σ is a symplectic involution on A. If (B, „ ) is a central simple algebra

with symplectic involution and char F = 2, then disc(σ — „ ) = 1. (If char F = 2,

(??) shows that σ — „ is symplectic.)

(6) Suppose σ is an orthogonal involution on A. If σ is hyperbolic, then disc σ =

1. (Since hyperbolic involutions in characteristic 2 are symplectic or unitary, the

hypotheses imply char F = 2.)

Proof : (??) If Int(u) —¦ σ is an orthogonal involution, then σ(u) = u and

Alt A, Int(u) —¦ σ = u · Alt(A, σ)

by (??). The property readily follows.

(??) It su¬ces to show that u ∈ Alt A, Int(u) —¦ σ . This is clear if char F = 2,

since the condition that σ is symplectic and Int(u) —¦ σ is orthogonal implies σ(u) =

’u, by (??). If char F = 2 we have Alt A, Int(u) —¦ σ = u · Alt(A, σ) by (??) and

1 ∈ Alt(A, σ) by (??).

(??) Let n = 2m = dim V and identify A with Mn (F ) by means of a basis e

of V . Let also be ∈ GLn (F ) be the Gram matrix of the bilinear form b with respect

to the chosen basis e. The involution σb is then given by

σb = Int(b’1 ) —¦ t,

e

82 II. INVARIANTS OF INVOLUTIONS

where t is the transpose involution. It is easily seen that disc t = (’1)m (indeed, it

su¬ces to ¬nd an alternating matrix of determinant 1), hence (??) yields:

disc σb = (’1)m det(b’1 ) · F —2 = disc b.

e

(??) If a ∈ Alt(A, σ) © A— , then a — 1 ∈ Alt(A — B, σ — „ ) © (A — B)— . The

property follows from the relation

NrdA—B (a — 1) = NrdA (a)deg B .

(??) Since „ is symplectic, deg B is even, by (??). The same argument as

in (??) applies to yield a — 1 ∈ Alt(A — B, σ — „ ) satisfying

NrdA—B (a — 1) ∈ F —2 .

(??) Let deg A = 2m and let e ∈ A be an idempotent such that e + σ(e) = 1.

We have rdim(eA) = m, hence, over a splitting ¬eld, e may be represented by a

diagonal matrix

e = diag(1, . . . , 1, 0, . . . , 0).

m m

Since σ(e) = 1 ’ e, we have 2e ’ 1 ∈ Alt(A, σ); on the other hand, over a splitting

¬eld,

2e ’ 1 = diag(1, . . . , 1, ’1, . . . , ’1),

m m

hence NrdA (2e ’ 1) = (’1)m and therefore disc σ = 1.

(7.4) Example. Let Q be a quaternion algebra with canonical involution γ. By

Proposition (??), every orthogonal involution on Q has the form σ = Int(s) —¦ γ

for some invertible s ∈ Skew(Q, γ) F . Proposition (??) shows that disc σ =

’ NrdQ (s) · F —2 . Therefore, if two orthogonal involutions σ = Int(s) —¦ γ and

σ = Int(s ) —¦ γ have the same discriminant, then we may assume that s and s have

the same reduced norm, hence also the same reduced characteristic polynomial

since TrdQ (s) = 0 = TrdQ (s ). Therefore,

s = xsx’1 = NrdQ (x)’1 xsγ(x)

for some x ∈ Q— , and it follows that

σ = Int(x) —¦ σ —¦ Int(x)’1 .

This show that orthogonal involutions on a quaternion algebra are classi¬ed up to

conjugation by their discriminant.

Observe also that if σ = Int(s) —¦ γ has trivial discriminant, then s2 ∈ F —2 .

Since s ∈ F , this relation implies that Q splits, hence quaternion division algebras

do not carry any orthogonal involution with trivial discriminant.

The next proposition may be seen as an analogue of the formula for the deter-

minant of an orthogonal sum of two bilinear spaces. Let (A, σ) be a central simple

F -algebra with orthogonal involution and let e1 , e2 ∈ A be symmetric idempotents

such that e1 + e2 = 1. Denote A1 = e1 Ae1 and A2 = e2 Ae2 . These algebras are

central simple and Brauer-equivalent to A (see (??)). They are not subalgebras of

A, however, since their unit elements e1 and e2 are not the unit 1 of A. The involu-

tion σ restricts to involutions σ1 and σ2 on A1 and A2 . If (A, σ) = EndF (V ), σb

for some vector space V and some nonsingular symmetric, nonalternating, bilinear

form b, then e1 and e2 are the orthogonal projections on some subspaces V1 , V2

§7. THE DISCRIMINANT 83

⊥

such that V = V1 • V2 . The algebras A1 , A2 may be identi¬ed with EndF (V1 ) and

EndF (V2 ), and σ1 , σ2 are the adjoint involutions with respect to the restrictions of

b to V1 and V2 . These restrictions clearly are symmetric, but if char F = 2 one of

them may be alternating. Therefore, in the general case, extension of scalars to a

splitting ¬eld of A shows that σ1 and σ2 are both orthogonal if char F = 2, but one

of them may be symplectic if char F = 2.

(7.5) Proposition. With the notation above,

det σ = det σ1 det σ2

where we set det σi = 1 if σi is symplectic.

Proof : Let ai ∈ Alt(Ai , σi ) for i = 1, 2; then a1 + a2 ∈ Alt(A, σ), and scalar

extension to a splitting ¬eld of A shows that

NrdA (a1 + a2 ) = NrdA1 (a1 ) NrdA2 (a2 ).

This completes the proof, since NrdAi (ai ) · F —2 = det σi if σi is orthogonal, and

NrdAi (ai ) ∈ F —2 if σi is symplectic, by (??).

7.B. The discriminant of quadratic pairs. Let (σ, f ) be a quadratic pair

on a central simple F -algebra of even degree. If char F = 2, the involution σ is

orthogonal and the map f is the restriction of 1 TrdA to Sym(A, σ); we then set

2

det(σ, f ) = det σ ∈ F — /F —2 disc(σ, f ) = disc σ ∈ F — /F —2 ;

and

this is consistent with the property that the discriminant of a quadratic form of

even dimension is equal to the discriminant of its polar bilinear form.

For the rest of this subsection, assume char F = 2. Recall that we write

SrdA : A ’ F for the map which associates to every element in A the coe¬cient

of X deg A’2 in its reduced characteristic polynomial (see (??)). Recall also that

„˜(x) = x2 + x for x ∈ F .

(7.6) Proposition. Let ∈ A be such that f (s) = TrdA ( s) for all s ∈ Sym(A, σ)

(see (??)). For all x ∈ A,

SrdA + x + σ(x) = SrdA ( ) + „˜ TrdA (x) .

Proof : It su¬ces to prove this formula after scalar extension to a splitting ¬eld. We

may therefore assume A = Mn (F ). By (??), we may ¬nd an element a ∈ Mn (F )

such that a + at ∈ GLn (F ) and (σ, f ) = (σa , fa ). Letting g = a + at , we then have

f (s) = tr(g ’1 as) for all s ∈ Sym(A, σ). Since (??) shows that the element is

uniquely determined up to the addition of an element in Alt(A, σ), it follows that

= g ’1 a + m + σ(m) for some m ∈ A,

hence

+ x + σ(x) = g ’1 a + (m + x) + σ(m + x).

Since σ = σa = Int(g ’1 ) —¦ t and g t = g, the right side may be rewritten as

g ’1 a + g ’1 y + g ’1 y t , where y = g(m + x).

As proved in (??), we have s2 (g ’1 a + g ’1 y + g ’1 y t ) = s2 (g ’1 a) + „˜ tr(g ’1 y) ,

hence

+ x + σ(x) = s2 (g ’1 a) + „˜ tr(m + x) = s2 (g ’1 a) + „˜ tr(m) + „˜ tr(x) .

s2

84 II. INVARIANTS OF INVOLUTIONS

In particular, by letting x = 0 we obtain s2 ( ) = s2 (g ’1 a) + „˜ tr(m) , hence the

preceding relation yields

s2 + x + „˜(x) = s2 ( ) + „˜ tr(x) .

(7.7) De¬nition. Let (σ, f ) be a quadratic pair on a central simple algebra A

over a ¬eld F of characteristic 2. By (??), there exists an element ∈ A such that

f (s) = TrdA ( s) for all s ∈ Sym(A, σ), and this element is uniquely determined up

to the addition of an element in Alt(A, σ). The preceding proposition shows that

the element SrdA ( ) + „˜(F ) ∈ F/„˜(F ) does not depend on the choice of ; we may

therefore set

det(σ, f ) = SrdA ( ) + „˜(F ) ∈ F/„˜(F )

and, letting deg A = 2m,

m(m’1)

disc(σ, f ) = det(σ, f ) + ∈ F/„˜(F ).

2

The following proposition justi¬es the de¬nitions above:

(7.8) Proposition. Let (V, q) be a nonsingular quadratic space of even dimension

over an arbitrary ¬eld F . For the associated quadratic pair (σ q , fq ) on EndF (V )

de¬ned in (??),

disc(σq , fq ) = disc q.

Proof : If char F = 2, the proposition follows from (??). For the rest of the proof

we therefore assume that char F = 2. Let dim V = n = 2m and consider a basis

(e1 , . . . , en ) of V which is symplectic for the polar form bq , i.e.,

bq (e2i’1 , e2i ) = 1, bq (e2i , e2i+1 ) = 0 and bq (ei , ej ) = 0 if |i ’ j| > 1.

As observed in (??), an element ∈ EndF (V ) such that fq (s) = Trd( s) for all

s ∈ Sym EndF (V ), σq is given by

m

= •q e2i’1 — e2i’1 q(e2i ) + e2i — e2i q(e2i’1 ) + e2i’1 — e2i

i=1

∼

where •q : V — V ’ EndF (V ) is the standard identi¬cation (??) associated to bq .

’

Furthermore we have, by (??), (??),

m m(m’1)

s2 ( ) = q(e2i’1 )q(e2i ) + ,

i=1 2

and therefore

m

disc(σq , fq ) = q(e2i’1 )q(e2i ) = disc q.

i=1

(7.9) Proposition. The discriminant of any hyperbolic quadratic pair is trivial.

Proof : If char F = 2, the proposition follows from (??). We may thus assume

that char F = 2. Let (σ, f ) be a hyperbolic quadratic pair on a central simple

F -algebra A. By (??), there is an idempotent e such that f (s) = TrdA (es) for all

s ∈ Sym(A, σ); thus

m(m’1)

disc(σ, f ) = SrdA (e) + + „˜(F )

2

where m = 1 deg A. Since e is an idempotent such that rdim(eA) = m, we have

2

PrdA,e (X) = (X ’ 1)m , hence SrdA (e) = m , and therefore disc(σ, f ) = 0.

2

§7. THE DISCRIMINANT 85

The discriminant of the tensor product of a quadratic pair with an involution

is calculated in the next proposition. We consider only the case where char F = 2,

since the case of characteristic di¬erent from 2 reduces to the tensor product of

involutions discussed in (??).

(7.10) Proposition. Suppose char F = 2. Let (σ1 , f1 ) be an orthogonal pair on a

central simple F -algebra A1 of degree n1 = 2m1 and let (A2 , σ2 ) be a central simple

F -algebra with involution of the ¬rst kind, of degree n2 . The determinant and the

discriminant of the orthogonal pair (σ1 — σ2 , f1— ) on A1 — A2 de¬ned in (??) are

as follows:

n2

det(σ1 — σ2 , f1— ) = n2 det(σ1 , f1 ) + m1 ;

2

disc(σ1 — σ2 , f1— ) = n2 disc(σ1 , f1 ).

In particular, if σ2 is symplectic, the discriminant of the canonical quadratic pair

(σ1 — σ2 , f— ) on A1 — A2 is trivial since n2 is even.

Proof : Let 1 ∈ A1 be such that f1 (s1 ) = TrdA1 ( 1 s1 ) for all s1 ∈ Sym(A1 , σ1 ).

We claim that the element = 1 — 1 satis¬es f1— (s) = TrdA1 —A2 ( s) for all s ∈

Sym(A1 — A2 , σ1 — σ2 ). By (??), we have

Sym(A1 — A2 , σ1 — σ2 ) = Symd(A1 — A2 , σ1 — σ2 ) + Sym(A1 , σ1 ) — Sym(A2 , σ2 ),

hence it su¬ces to show