Proposition (??) then yields σ = Int(u) —¦ t for some invertible alternating matrix

u ∈ A— , hence Symd(A, σ) = u · Alt Mn (F ), t . Therefore, there exists a matrix

a ∈ Alt Mn (F ), t such that s = ua. The (reduced) characteristic polynomial of s

is then

PrdA,s (X) = det(X · 1 ’ s) = (det u) det(X · u’1 ’ a) .

Since u and X · u’1 ’ a are alternating, their determinants are the squares of their

pfa¬an pf u, pf(X · u’1 ’ a) (see for instance E. Artin [?, Theorem 3.27]), hence

PrdA,s (X) = [(pf u) pf(X · u’1 ’ a)]2 .

Let deg A = n = 2m. In view of the preceding proposition, for every s ∈

Symd(A, σ) there is a unique monic polynomial, the pfa¬an characteristic polyno-

mial, Prpσ,s (X) ∈ F [X] of degree m such that

PrdA,s (X) = Prpσ,s (X)2 .

For s ∈ Symd(A, σ), we de¬ne the pfa¬an trace Trpσ (s) and the pfa¬an norm

Nrpσ (s) ∈ F as coe¬cients of Prpσ,s (X):

Prpσ,s (X) = X m ’ Trpσ (s)X m’1 + · · · + (’1)m Nrpσ (s).

(2.10)

20 I. INVOLUTIONS AND HERMITIAN FORMS

Thus, Trpσ and Nrpσ are polynomial maps of degree 1 and m respectively on

Symd(A, σ), and we have

NrdA (s) = Nrpσ (s)2

(2.11) TrdA (s) = 2 Trpσ (s) and

for all s ∈ Symd(A, σ). Moreover, we have PrdA,1 (X) = (X ’ 1)2m , hence

Prpσ,1 (X) = (X ’ 1)m and therefore

(2.12) Trpσ (1) = m and Nrpσ (1) = 1.

Since polynomial maps on Symd(A, σ) form a domain, the map Nrpσ is uniquely

determined by (??) and (??). (Of course, if char F = 2, the map Trpσ is also

uniquely determined by (??); in all characteristics it is uniquely determined by the

property in (??) below.) Note that the pfa¬an norm Nrpσ (or simply pfa¬an) is

an analogue of the classical pfa¬an. However it is de¬ned on the space Symd(A, σ)

whereas pf is de¬ned on alternating matrices (under the transpose involution).

Nevertheless, it shares with the pfa¬an the fundamental property demonstrated in

the following proposition:

(2.13) Proposition. For all s ∈ Symd(A, σ) and all a ∈ A,

Trpσ σ(a) + a = TrdA (a) Nrpσ σ(a)sa = NrdA (a) Nrpσ (s).

and

Proof : We ¬rst prove the second equation. If s is not invertible, then NrdA (s) =

NrdA σ(a)sa = 0, hence Nrpσ (s) = Nrpσ σ(a)sa = 0, proving the equation in

this particular case. For s ∈ Symd(A, σ) © A— ¬xed, consider both sides of the

equality to be proved as polynomial maps on A:

f1 : a ’ Nrpσ σ(a)sa and f2 : a ’ NrdA (a) Nrpσ (s).

Since NrdA σ(a)sa = NrdA (a)2 NrdA (s), we have f1 = f2 , hence (f1 + f2 )(f1 ’

2 2

f2 ) = 0. Since polynomial maps on A form a domain, it follows that f1 = ±f2 .

Taking into account the fact that f1 (1) = Nrpσ (s) = f2 (1), we get f1 = f2 .

The ¬rst equation follows from the second. For, let t be an indeterminate over

F and consider the element 1 + ta ∈ AF (t) . By the equation just proven, we have

1 ’ tσ(a) (1 ’ ta) = NrdA (1 ’ ta) = 1 ’ TrdA (a)t + SrdA (a)t2 ’ . . .

Nrpσ

On the other hand, for all s ∈ Symd(A, σ) we have

Nrpσ (1 ’ s) = Prpσ,s (1) = 1 ’ Trpσ (s) + · · · + (’1)m Nrpσ (s),

hence the coe¬cient of ’t in Nrpσ 1 ’ tσ(a) (1 ’ ta) is Trpσ σ(a) + a ; the ¬rst

equality is thus proved.

2.B. Involutions of the second kind. In the case of involutions of the

second kind on a simple algebra B, the base ¬eld F is usually not the center of the

algebra, but the sub¬eld of central invariant elements which is of codimension 2

in the center. Under scalar extension to an algebraic closure of F , the algebra B

decomposes into a direct product of two simple factors. It is therefore convenient

to extend our discussion of involutions of the second kind to semisimple F -algebras

of the form B1 — B2 , where B1 , B2 are central simple F -algebras.

Throughout this section, we thus denote by B a ¬nite dimensional F -algebra

whose center K is a quadratic ´tale3 extension of F , and assume that B is either

e

simple (if K is a ¬eld) or a direct product of two simple algebras (if K F — F ).

3 Thissimply means that K is either a ¬eld which is a separable quadratic extension of F , or

F — F . See § ?? for more on ´tale algebras.

e

K

§2. INVOLUTIONS 21

We denote by ι the nontrivial automorphism of K/F and by „ an involution of the

second kind on B, whose restriction to K is ι. For convenience, we refer to (B, „ )

as a central simple F -algebra with involution of the second kind, even though

its center is not F and the algebra B may not be simple.4 A homomorphism

f : (B, „ ) ’ (B , „ ) is then an F -algebra homomorphism f : B ’ B such that

„ —¦ f = f —¦ „.

If L is any ¬eld containing F , the L-algebra BL = B —F L has center KL =

K —F L, a quadratic ´tale extension of L, and carries an involution of the second

e

kind „L = „ — IdL . Moreover, (BL , „L ) is a central simple L-algebra with involution

of the second kind.

As a parallel to the terminology of types used for involutions of the ¬rst kind,

and because of their relation with unitary groups (see Chapter ??), involutions of

the second kind are also called of unitary type (or simply unitary).

We ¬rst examine the case where the center K is not a ¬eld.

(2.14) Proposition. If K F — F , there is a central simple F -algebra E such

that

(E — E op , µ),

(B, „ )

where the involution µ is de¬ned by µ(x, y op ) = (y, xop ). This involution is called

the exchange involution.

Proof : Let B = B1 — B2 where B1 , B2 are central simple F -algebras. Since the

restriction of „ to the center K = F — F interchanges the two factors, it maps (1, 0)

to (0, 1), hence

„ B1 — {0} = „ (B1 — B2 ) · (1, 0) = (0, 1) · (B1 — B2 ) = {0} — B2 .

It follows that B1 and B2 are anti-isomorphic. We may then de¬ne an F -algebra

op ∼

isomorphism f : B1 ’ B2 by the relation

’

„ (x, 0) = 0, f (xop ) ,

op

and identify B1 — B2 with B1 — B1 by mapping (x1 , x2 ) to x1 , f ’1 (x2 ) . Under

this map, „ is identi¬ed with the involution µ.

In view of this proposition, we may de¬ne the degree of the central simple

F -algebra (B, „ ) with involution of the second kind by

deg B if K is a ¬eld,

deg(B, „ ) =

(E — E op , µ).

deg E if K F — F and (B, „ )

2

Equivalently, deg(B, „ ) is de¬ned by the relation dimF B = 2 deg(B, „ ) .

If the center K of B is a ¬eld, (??) applies to BK = B —F K, since its center

is KK = K —F K K — K. In this case we get a canonical isomorphism:

(2.15) Proposition. Suppose that the center K of B is a ¬eld. There is a canon-

ical isomorphism of K-algebras with involution

∼

• : (BK , „K ) ’ (B — B op , µ)

’

op

which maps b — ± to b±, „ (b)± for b ∈ B and ± ∈ K.

4 We thus follow Jacobson™s convention in [?, p. 208]; it can be justi¬ed by showing that (B, „ )

is indeed central simple as an algebra-with-involution.

22 I. INVOLUTIONS AND HERMITIAN FORMS

Proof : It is straightforward to check that • is a homomorphism of central simple

K-algebras with involution of the second kind. It thus su¬ces to prove that • has

an inverse. Let ± ∈ K F . Then the map Ψ : B — B op ’ BK de¬ned by

„ (y)± ’ xι(±) x ’ „ (y)

Ψ(x, y op ) = —1+ —±

± ’ ι(±) ± ’ ι(±)

is the inverse of •.

In a semisimple algebra of the form B1 — B2 , where B1 , B2 are central sim-

ple F -algebras of the same degree, the reduced characteristic polynomial of an

element (b1 , b2 ) may be de¬ned as PrdB1 ,b1 (X), PrdB2 ,b2 (X) ∈ (F — F )[X],

where PrdB1 ,b1 (X) and PrdB2 ,b2 (X) are the reduced characteristic polynomials of

b1 and b2 respectively (see Reiner [?, p. 121]). Since the reduced characteristic

polynomial of an element does not change under scalar extension (see Reiner [?,

Theorem (9.27)]), the preceding proposition yields:

(2.16) Corollary. For every b ∈ B, the reduced characteristic polynomials of b

and „ (b) are related by

PrdB,„ (b) = ι(PrdB,b ) in K[X].

In particular, TrdB „ (b) = ι TrdB (b) and NrdB „ (b) = ι NrdB (b) .

Proof : If K F — F , the result follows from (??); if K is a ¬eld, it follows

from (??).

As for involutions of the ¬rst kind, we may de¬ne the sets of symmetric, skew-

symmetric, symmetrized and alternating elements in (B, „ ) by

Sym(B, „ ) = { b ∈ B | „ (b) = b },

Skew(B, „ ) = { b ∈ B | „ (b) = ’b },

Symd(B, „ ) = { b + „ (b) | b ∈ B },

Alt(B, „ ) = { b ’ „ (b) | b ∈ B }.

These sets are vector spaces over F . In contrast with the case of involutions of the

¬rst kind, there is a straightforward relation between symmetric, skew-symmetric

and alternating elements, as the following proposition shows:

(2.17) Proposition. Symd(B, „ ) = Sym(B, „ ) and Alt(B, „ ) = Skew(B, „ ). For

any ± ∈ K — such that „ (±) = ’±,

Skew(B, „ ) = ± · Sym(B, „ ).

If deg(B, „ ) = n, then

dimF Sym(B, „ ) = dimF Skew(B, „ ) = dimF Symd(B, „ ) = dimF Alt(B, „ ) = n2 .

Proof : The relations Skew(B, „ ) = ± · Sym(B, „ ) and Symd(B, „ ) ‚ Sym(B, „ ),

Alt(B, „ ) ‚ Skew(B, „ ) are clear. If β ∈ K is such that β + ι(β) = 1, then every

element s ∈ Symd(B, „ ) may be written as s = βs + „ (βs), hence Sym(B, „ ) =

Symd(B, „ ). Similarly, every element s ∈ Skew(B, „ ) may be written as s = βs ’

„ (βs), hence Skew(B, „ ) = Alt(B, „ ). Therefore, the vector spaces Sym(B, „ ),

Skew(B, „ ), Symd(B, „ ) and Alt(B, „ ) have the same dimension. This dimension

1

is 2 dimF B, since Alt(B, „ ) is the image of the F -linear endomorphism Id ’ „

of B, whose kernel is Sym(B, „ ). Since dimF B = 2 dimK B = 2n2 , the proof is

complete.

§2. INVOLUTIONS 23

As for involutions of the ¬rst kind, all the involutions of the second kind on B

which have the same restriction to K as „ are obtained by composing „ with an

inner automorphism, as we now show.

(2.18) Proposition. Let (B, „ ) be a central simple F -algebra with involution of

the second kind, and let K denote the center of B.

(1) For every unit u ∈ B — such that „ (u) = »u with » ∈ K — , the map Int(u) —¦ „ is

an involution of the second kind on B.

(2) Conversely, for every involution „ on B whose restriction to K is ι, there

exists some u ∈ B — , uniquely determined up to a factor in F — , such that

„ = Int(u) —¦ „ „ (u) = u.

and

In this case,

Sym(B, „ ) = u · Sym(B, „ ) = Sym(B, „ ) · u’1 .

2

Proof : Computation shows that Int(u) —¦ „ = Int u„ (u)’1 , and (??) follows.

If „ is an involution on B which has the same restriction to K as „ , the

composition „ —¦ „ is an automorphism which leaves K elementwise invariant. The

Skolem-Noether theorem shows that „ —¦ „ = Int(u0 ) for some u0 ∈ B — , hence

2

„ = Int(u0 ) —¦ „ . Since „ = Id, we have u0 „ (u0 )’1 ∈ K — . Let » ∈ K — be such

that „ (u0 ) = »u0 . Applying „ to both sides of this relation, we get NK/F (») = 1.

Hilbert™s theorem 90 (see (??)) yields an element µ ∈ K — such that » = µι(µ)’1 .

Explicitly one can take µ = ± + »ι(±) for ± ∈ K such that ± + »ι(±) is invertible.

The element u = µu0 then satis¬es the required conditions.

2.C. Examples.

Endomorphism algebras. Let V be a ¬nite dimensional vector space over

a ¬eld F . The involutions of the ¬rst kind on EndF (V ) have been determined

in the introduction to this chapter: every such involution is the adjoint involution

with respect to some nonsingular symmetric or skew-symmetric bilinear form on V ,

uniquely determined up to a scalar factor. Moreover, it is clear from De¬nition (??)

that the involution is orthogonal (resp. symplectic) if the corresponding bilinear

form is symmetric and nonalternating (resp. alternating).

Involutions of the second kind can be described similarly. Suppose that V

is a ¬nite dimensional vector space over a ¬eld K which is a separable quadratic

extension of some sub¬eld F with nontrivial automorphism ι. A hermitian form

on V (with respect to ι) is a bi-additive map

h: V — V ’ K

such that

h(v±, wβ) = ι(±)h(v, w)β for v, w ∈ V and ±, β ∈ K

and

h(w, v) = ι h(v, w) for v, w ∈ V .

The form h is called nonsingular if the only element x ∈ V such that h(x, y) = 0

for all y ∈ V is x = 0. If this condition holds, an involution σh on EndK (V ) may

be de¬ned by the following condition:

h x, f (y) = h σh (f )(x), y for f ∈ EndK (V ), x, y ∈ V .

24 I. INVOLUTIONS AND HERMITIAN FORMS

The involution σh on EndK (V ) is the adjoint involution with respect to the hermit-

ian form h. From the de¬nition of σh , it follows that σh (±) = ι(±) for all ± ∈ K,

hence σh is of the second kind. As for involutions of the ¬rst kind, one can prove

that every involution „ of the second kind on EndK (V ) whose restriction to K is

ι is the adjoint involution with respect to some nonsingular hermitian form on V ,

uniquely determined up to a factor in F — .

We omit the details of the proof, since a more general statement will be proved

in §?? below (see (??)).

Matrix algebras. The preceding discussion can of course be translated to

matrix algebras, since the choice of a basis in an n-dimensional vector space V

over F yields an isomorphism EndF (V ) Mn (F ). However, matrix algebras are

endowed with a canonical involution of the ¬rst kind, namely the transpose involu-

tion t. Therefore, a complete description of involutions of the ¬rst kind on Mn (F )

can also be easily derived from (??).

(2.19) Proposition. Every involution of the ¬rst kind σ on Mn (F ) is of the form

σ = Int(u) —¦ t

for some u ∈ GLn (F ), uniquely determined up to a factor in F — , such that ut = ±u.

Moreover, the involution σ is orthogonal if ut = u and u ∈ Alt Mn (F ), t , and it

is symplectic if u ∈ Alt Mn (F ), t .

If Mn (F ) is identi¬ed with EndF (F n ), the involution σ = Int(u) —¦ t is the

adjoint involution with respect to the nonsingular form b on F n de¬ned by

b(x, y) = xt · u’1 · y for x, y ∈ F n .

Suppose now that A is an arbitrary central simple algebra over a ¬eld F and

that is an involution (of any kind) on A. We de¬ne an involution — on Mn (A) by

(aij )— t

1¤i,j¤n = (aij )1¤i,j¤n .

(2.20) Proposition. The involution — is of the same type as . Moreover, the

involutions σ on Mn (A) such that σ(±) = ± for all ± ∈ F can be described as

follows:

(1) If is of the ¬rst kind, then every involution of the ¬rst kind on Mn (A) is of

the form σ = Int(u) —¦ — for some u ∈ GLn (A), uniquely determined up to a factor

in F — , such that u— = ±u. If char F = 2, the involution Int(u) —¦ — is of the same

type as if and only if u— = u. If char F = 2, the involution Int(u) —¦ — is symplectic

if and only if u ∈ Alt Mn (A), — .

(2) If is of the second kind, then every involution of the second kind σ on M n (A)

such that σ(±) = ± for all ± ∈ F is of the form σ = Int(u)—¦ — for some u ∈ GLn (A),

uniquely determined up to a factor in F — invariant under , such that u— = u.

Proof : From the de¬nition of — it follows that ±— = ± for all ± ∈ F . Therefore, the

involutions — and are of the same kind.

Suppose that is of the ¬rst kind. A matrix (aij )1¤i,j¤n is — -symmetric if and

only if its diagonal entries are -symmetric and aji = aij for i > j, hence

n(n’1)

dim Sym Mn (A), — = n dim Sym(A, ) + dim A.

2

If deg A = d and dim Sym(A, ) = d(d + δ)/2, where δ = ±1, we thus get

dim Sym Mn (A), — = nd(nd + δ)/2.

§2. INVOLUTIONS 25

Therefore, if char F = 2 the type of — is the same as the type of .

n

Since TrdMn (A) (aij )1¤i,j¤n = i=1 TrdA (aii ), we have

TrdMn (A) Sym Mn (A), — = {0} if and only if TrdA Sym(A, ) = {0}.

Therefore, when char F = 2 the involution — is symplectic if and only if is sym-

plectic.

We have thus shown that in all cases the involutions — and are of the same

type (orthogonal, symplectic or unitary). The other assertions follow from (??)

and (??).

In §?? below, it is shown how the various involutions on Mn (A) are associated

to hermitian forms on An under the identi¬cation EndA (An ) = Mn (A).

Quaternion algebras. A quaternion algebra over a ¬eld F is a central simple

F -algebra of degree 2. If the characteristic of F is di¬erent from 2, it can be shown

(see Scharlau [?, §8.11]) that every quaternion algebra Q has a basis (1, i, j, k)

subject to the relations

i2 ∈ F — , j 2 ∈ F —, ij = k = ’ji.

Such a basis is called a quaternion basis; if i2 = a and j 2 = b, the quaternion

algebra Q is denoted

Q = (a, b)F .

Conversely, for any a, b ∈ F — the 4-dimensional F -algebra Q with basis (1, i, j, k)

where multiplication is de¬ned through the relations i2 = a, j 2 = b, ij = k = ’ji

is central simple and is therefore a quaternion algebra (a, b)F .

If char F = 2, every quaternion algebra Q has a basis (1, u, v, w) subject to the

relations

u2 + u ∈ F, v2 ∈ F —, uv = w = vu + v

(see Draxl [?, §11]). Such a basis is called a quaternion basis in characteristic 2. If

u2 + u = a and v 2 = b, the quaternion algebra Q is denoted

Q = [a, b)F .

Conversely, for all a ∈ F , b ∈ F — , the relations u2 + u = a, v 2 = b and vu = uv + v

give the span of 1, u, v, uv the structure of a quaternion algebra.

Quaternion algebras in characteristic 2 may alternately be de¬ned as algebras

generated by two elements r, s subject to

r2 ∈ F, s2 ∈ F, rs + sr = 1.

Indeed, if r2 = 0 the algebra thus de¬ned is isomorphic to M2 (F ); if r2 = 0 it

has a quaternion basis (1, sr, r, sr 2 ). Conversely, every quaternion algebra with

quaternion basis (1, u, v, w) as above is generated by r = v and s = uv ’1 satisfying

the required relations. The quaternion algebra Q generated by r, s subject to the

relations r2 = a ∈ F , s2 = b ∈ F and rs + sr = 1 is denoted

Q = a, b .

F

Thus, a, b F M2 (F ) if a (or b) = 0 and a, b F [ab, a)F if a = 0. The

quaternion algebra a, b F is thus the Cli¬ord algebra of the quadratic form [a, b].

For every quaternion algebra Q, an F -linear map γ : Q ’ Q can be de¬ned by

γ(x) = TrdQ (x) ’ x for x ∈ Q

26 I. INVOLUTIONS AND HERMITIAN FORMS

where TrdQ is the reduced trace in Q. Explicitly, for x0 , x1 , x2 , x3 ∈ F ,

γ(x0 + x1 i + x2 j + x3 k) = x0 ’ x1 i ’ x2 j ’ x3 k

if char F = 2 and

γ(x0 + x1 u + x2 v + x3 w) = x0 + x1 (u + 1) + x2 v + x3 w

if char F = 2. For the split quaternion algebra Q = M2 (F ) (in arbitrary character-

istic),

x11 x12 x22 ’x12

γ =

x21 x22 ’x21 x11

Direct computations show that γ is an involution, called the quaternion con-

jugation or the canonical involution. If char F = 2, then Sym(Q, γ) = F and

Skew(Q, γ) has dimension 3. If char F = 2, then Sym(Q, γ) is spanned by 1, v, w

which have reduced trace equal to zero. Therefore, the involution γ is symplectic

in every characteristic.

(2.21) Proposition. The canonical involution γ on a quaternion algebra Q is the

unique symplectic involution on Q. Every orthogonal involution σ on Q is of the

form

σ = Int(u) —¦ γ

where u is an invertible quaternion in Skew(Q, γ) F which is uniquely determined

by σ up to a factor in F — .

Proof : From (??), it follows that every involution of the ¬rst kind σ on Q has the

form σ = Int(u) —¦ γ where u is a unit such that γ(u) = ±u. Suppose that σ is

symplectic. If char F = 2, Proposition (??) shows that u ∈ Alt(Q, γ) = F , hence

σ = γ. Similarly, if char F = 2, Proposition (??) shows that γ(u) = u, hence

u ∈ F — and σ = γ.

Unitary involutions on quaternion algebras also have a very particular type, as

we proceed to show.

(2.22) Proposition (Albert). Let K/F be a separable quadratic ¬eld extension

with nontrivial automorphism ι. Let „ be an involution of the second kind on a

quaternion algebra Q over K, whose restriction to K is ι. There exists a unique

quaternion F -subalgebra Q0 ‚ Q such that

Q = Q 0 —F K „ = γ0 — ι

and

where γ0 is the canonical involution on Q0 . Moreover, the algebra Q0 is uniquely

determined by these conditions.

Proof : Let γ be the canonical involution on Q. Then „ —¦ γ —¦ „ is an involution

of the ¬rst kind and symplectic type on Q, so „ —¦ γ —¦ „ = γ by (??). From this

last relation, it follows that „ —¦ γ is a ι-semilinear automorphism of order 2 of Q.

The F -subalgebra Q0 of invariant elements then satis¬es the required conditions.

Since these conditions imply that every element in Q0 is invariant under „ —¦ γ, the

algebra Q0 is uniquely determined by „ . (It is also the F -subalgebra of Q generated

by „ -skew-symmetric elements of trace zero, see (??).)

The proof holds without change in the case where K F — F , provided that

quaternion algebras over K are de¬ned as direct products of two quaternion F -

algebras.

§2. INVOLUTIONS 27

Symbol algebras. Let n be an arbitrary positive integer and let K be a ¬eld

containing a primitive nth root of unity ζ. For a, b ∈ K — , write (a, b)ζ,K for the

K-algebra generated by two elements x, y subject to the relations

xn = a, y n = b, yx = ζxy.

This algebra is central simple of degree n (see Draxl [?, §11]); it is called a symbol

algebra.5 Clearly, quaternion algebras are the symbol algebras of degree 2.

If K has an automorphism ι of order 2 which leaves a and b invariant and maps

ζ to ζ ’1 , then this automorphism extends to an involution „ on (a, b)ζ,K which

leaves x and y invariant.

Similarly, any automorphism ι of order 2 of K which leaves a and ζ invariant

and maps b to b’1 (if any) extends to an involution σ on (a, b)ζ,K which leaves x

invariant and maps y to y ’1 .

Tensor products.

(2.23) Proposition. (1) Let (A1 , σ1 ), . . . , (An , σn ) be central simple F -algebras

with involution of the ¬rst kind. Then σ1 — · · · — σn is an involution of the ¬rst

kind on A1 —F · · · —F An . If char F = 2, this involution is symplectic if and only if

an odd number of involutions among σ1 , . . . , σn are symplectic. If char F = 2, it

is symplectic if and only if at least one of σ1 , . . . , σn is symplectic.

(2) Let K/F be a separable quadratic ¬eld extension with nontrivial automorphism ι

and let (B1 , „1 ), . . . , (Bn , „n ) be central simple F -algebras with involution of the

second kind with center K. Then „1 — · · · — „n is an involution of the second kind

on B1 —K · · · —K Bn .

(3) Let K/F be a separable quadratic ¬eld extension with nontrivial automorphism ι

and let (A, σ) be a central simple F -algebra with involution of the ¬rst kind. Then

(A —F K, σ — ι) is a central simple F -algebra with involution of the second kind.

The proof, by induction on n for the ¬rst two parts, is straightforward. In

case (??), we denote

(A1 , σ1 ) —F · · · —F (An , σn ) = (A1 —F · · · —F An , σ1 — · · · — σn ),

and use similar notations in the other two cases.

Tensor products of quaternion algebras thus yield examples of central simple

algebras with involution. Merkurjev™s theorem [?] shows that every central simple

algebra with involution is Brauer-equivalent to a tensor product of quaternion alge-

bras. However, there are examples of division algebras with involution of degree 8

which do not decompose into tensor products of quaternion algebras, and there are

examples of involutions σ on tensor products of two quaternion algebras which are