Therefore, the adjoint involutions σh and σh are related by

σh = Int(v) —¦ σh .

Therefore, if σh = σh , then v ∈ F — and the forms h, h di¬er by a factor in F — .

If θ is of the second kind and h, h are both hermitian, the relation h = v · h

implies that θ(v) = v. We have thus shown injectivity of the map h ’ σh on the

set of equivalence classes modulo factors in F — (invariant under θ) in both cases

(??) and (??).

Let D be a central division algebra Brauer-equivalent to E. We may then

identify E with Ms (D) for some integer s, hence also M with Mr,s (D) and A with

Mr (D), as in the proof of (??). We may thus assume that

A = Mr (D), M = Mr,s (D), E = Ms (D).

Theorem (??) shows that D carries an involution such that ± = θ(±) for all ± ∈ F .

We use the same notation — for the maps A ’ A, E ’ E and M ’ Ms,r (D) de¬ned

by

(dij )— = (dij )t .

i,j i,j

44 I. INVOLUTIONS AND HERMITIAN FORMS

Proposition (??) shows that the maps — on A and E are involutions of the same

type as .

Consider now case (??), where is of the ¬rst kind. According to (??), we may

¬nd u ∈ E — such that u— = ±u and θ = Int(u) —¦ — . Moreover, for any involution of

the ¬rst kind σ on A we may ¬nd some g ∈ A— such that g — = ±g and σ = Int(g)—¦ — .

De¬ne then a map h : M — M ’ E by

h(x, y) = u · x— · g ’1 · y for x, y ∈ M .

This map is clearly bi-additive. Moreover, for ±, β ∈ E and x, y ∈ M we have

h(x±, yβ) = u · ±— · x— · g ’1 · y · β = θ(±) · h(x, y) · β

and

—

h(y, x) = u · u— · x— · (g ’1 )— · y · u’1 = δθ h(x, y) ,

where δ = +1 if u’1 u— = g ’1 g — (= ±1) and δ = ’1 if u’1 u— = ’g ’1 g — (= 1).

Therefore, h is a hermitian or skew-hermitian form on M . For a ∈ A and x, y ∈ M ,

h(x, ay) = u · x— · (ga— g ’1 )— · g ’1 · y = h σ(a)x, y ,

hence σ is the adjoint involution with respect to h. To complete the proof of (??),

it remains to relate the type of σ to properties of h.

Suppose ¬rst that char F = 2. Proposition (??) shows that the type of θ (resp.

of σ) is the same as the type of if and only if u’1 u— = +1 (resp. g ’1 g — = +1).

Therefore, σ and θ are of the same type if and only if u’1 u— = g ’1 g — , and this

condition holds if and only if h is hermitian.

Suppose now that char F = 2. We have to show that h(x, x) ∈ Alt(E, θ) for

all x ∈ M if and only if σ is symplectic. Proposition (??) shows that this last

condition is equivalent to g ∈ Alt(A, — ). If g = a ’ a— for some a ∈ A, then

g ’1 = ’g ’1 g(g ’1 )— = b ’ b— for b = g ’1 a— (g ’1 )— . It follows that for all x ∈ M

h(x, x) = u · x— · b · x ’ θ(u · x— · b · x) ∈ Alt(E, θ).

Conversely, if h is alternating, then x— · g ’1 · x ∈ Alt(E, — ) for all x ∈ M , since (??)

shows that Alt(E, — ) = u’1 · Alt(E, θ). In particular, taking for x the matrix ei1

whose entry with indices (i, 1) is 1 and whose other entries are 0, it follows that the

i-th diagonal entry of g ’1 is in Alt(D, ). Let g ’1 = (gij )1¤i,j¤r and gii = di ’ di

for some di ∈ D. Then g ’1 = b ’ b— where the matrix b = (bij )1¤i,j¤r is de¬ned by

±

gij if i < j,

bij = di if i = j,

0 if i > j.

Therefore, g = ’gg ’1 g — = gb— g — ’ (gb— g — )— ∈ Alt(A, — ), completing the proof

of (??).

The proof of (??) is similar, but easier since there is only one type. Propo-

sition (??) yields an element u ∈ E — such that u— = u and θ = Int(u) —¦ — , and

shows that every involution σ on A such that σ(±) = ± for all ± ∈ F has the form

σ = Int(g) —¦ — for some g ∈ A— such that g — = g. The same computations as

for (??) show that σ is the adjoint involution with respect to the hermitian form h

on M de¬ned by

h(x, y) = u · x— · g ’1 · y for x, y ∈ M .

§4. HERMITIAN FORMS 45

The preceding theorem applies notably in the case where E is a division alge-

bra, to yield a correspondence between involutions on a central simple algebra A

and hermitian and skew-hermitian forms on vector spaces over the division algebra

Brauer-equivalent to A. However, (??) shows that a given central simple algebra

may be represented as A = EndE (M ) for any central simple algebra E Brauer-

equivalent to A (and for a suitable E-module M ). Involutions on A then corre-

spond to hermitian and skew-hermitian forms on M by the preceding theorem. In

particular, if A has an involution of the ¬rst kind, then a theorem of Merkurjev [?]

shows that we may take for E a tensor product of quaternion algebras.

4.B. Extension of involutions and transfer. This section analyzes the

possibility of extending an involution from a simple subalgebra. One type of exten-

sion is based on an analogue of the Scharlau transfer for hermitian forms which is

discussed next. The general extension result, due to Kneser, is given thereafter.

The transfer. Throughout this subsection, we consider the following situa-

tion: Z/F is a ¬nite extension of ¬elds, E is a central simple Z-algebra and T is a

central simple F -algebra contained in E. Let C be the centralizer of T in E. By

the double centralizer theorem (see (??)) this algebra is central simple over Z and

E = T —F C.

Suppose that θ is an involution on E (of any kind) which preserves T , hence also C.

For simplicity, we also call θ the restriction of θ to T and to C.

(4.3) De¬nition. An F -linear map s : E ’ T is called an involution trace if it

satis¬es the following conditions (see Knus [?, (7.2.4)]):

(1) s(t1 xt2 ) = t1 s(x)t2 for all x ∈ E and t1 , t2 ∈ T ;

(2) s θ(x) = θ s(x) for all x ∈ E;

(3) if x ∈ E is such that s θ(x)y = 0 for all y ∈ E, then x = 0.

In view of (??), condition (??) may equivalently be phrased as follows: the only

element y ∈ E such that s θ(x)y = 0 for all x ∈ E is y = 0. It is also equivalent

to the following:

(?? ) ker s does not contain any nontrivial left or right ideal in E.

Indeed, I is a right (resp. left) ideal in ker s if and only if s θ(x)y = 0 for all

θ(x) ∈ I and all y ∈ E (resp. for all x ∈ E and y ∈ I).

For instance, if T = F = Z, the reduced trace TrdE : E ’ F is an involution

trace. Indeed, condition (??) follows from (??) if θ is of the ¬rst kind and from (??)

if θ is of the second kind, and condition (??) follows from the fact that the bilinear

(reduced) trace form is nonsingular (see (??)).

If E = Z and T = F , every nonzero linear map s : Z ’ F which commutes

with θ is an involution trace. Indeed, if x ∈ Z is such that s θ(x)y = 0 for all

y ∈ Z, then x = 0 since s = 0 and Z = { θ(x)y | y ∈ Z } if x = 0.

The next proposition shows that every involution trace s : E ’ T can be ob-

tained by combining these particular cases.

(4.4) Proposition. Fix a nonzero linear map : Z ’ F which commutes with θ.

For every unit u ∈ Sym(C, θ), the map s : E ’ T de¬ned by

s(t — c) = t · TrdC (uc) for t ∈ T and c ∈ C

is an involution trace. Every involution trace from E to T is of the form above for

some unit u ∈ Sym(C, θ).

46 I. INVOLUTIONS AND HERMITIAN FORMS

Proof : Conditions (??) and (??) are clear. Suppose that x = i ti — ci ∈ E is

such that s θ(x)y = 0 for all y ∈ E. We may assume that the elements ti ∈ T

are linearly independent over F . The relation s θ(x) · 1 — c = 0 for all c ∈ C then

yields TrdC uθ(ci )c = 0 for all i and all c ∈ C. Since is nonzero, it follows

that TrdC uθ(ci )c = 0 for all i and all c ∈ C, hence uθ(ci ) = 0 for all i since the

bilinear reduced trace form is nonsingular. It follows that θ(ci ) = 0 for all i since u

is invertible, hence x = 0.

Let s : E ’ T be an arbitrary involution trace. For t ∈ T and c ∈ C,

t · s(1 — c) = s(t — c) = s(1 — c) · t,

hence the restriction of s to C takes values in F and s = IdT — s0 where s0 : C ’ F

denotes this restriction. Since is nonzero and the bilinear reduced trace form is

nonsingular, the linear map C ’ HomF (C, F ) which carries c ∈ C to the linear map

x ’ TrdC (cx) is injective, hence also surjective, by dimension count. Therefore,

there exists u ∈ C such that s0 (x) = TrdC (ux) for all x ∈ C. If u is not

invertible, then the annihilator of the left ideal generated by u is a nontrivial right

ideal in the kernel of s0 , contrary to the hypothesis that s is an involution trace.

Finally, observe that for c ∈ C,

s0 θ(c) = θ TrdC cθ(u) = TrdC θ(u)c ,

hence the condition s0 θ(c) = s0 (c) for all c ∈ C implies that θ(u) = u.

(4.5) Corollary. For every involution trace s : E ’ T , there exists an involution

θs on C such that

s θ(c)x = s xθs (c) for c ∈ C, x ∈ E.

The involutions θs and θ have the same restriction to Z.

Proof : Fix a nonzero linear map : Z ’ F which commutes with θ. According

to (??), we have s = IdT — s0 where s0 : C ’ F is de¬ned by s0 (c) = TrdC (uc)

for some symmetric unit u ∈ C — . Let θs = Int(u) —¦ θ. For c, c ∈ C,

TrdC uθ(c)c = TrdC θs (c)uc = TrdC uc θs (c) ,

hence for all t ∈ T ,

s θ(c) · (t — c ) = t · TrdC uθ(c)c = s (t — c ) · θs (c) .

Therefore, the involution θs satis¬es

s θ(c)x = s xθs (c) for c ∈ C, x ∈ E.

The involution θs is uniquely determined by this condition, because if s xθs (c) =

s xθs (c) for all c ∈ C and x ∈ E, then property (??) of involution traces in (??)

implies that θs (c) = θs (c) for all c ∈ C.

Since θs = Int(u) —¦ θ, it is clear that θs (z) = θ(z) for all z ∈ Z.

Using an involution trace s : E ’ T , we may de¬ne a structure of hermitian

module over T on every hermitian module over E, as we proceed to show.

Suppose that M is a ¬nitely generated right module over E. Since T ‚ E, we

may also consider M as a right T -module, and

EndE (M ) ‚ EndT (M ).

The centralizer of EndE (M ) in EndT (M ) is easily determined:

§4. HERMITIAN FORMS 47

(4.6) Lemma. For c in the centralizer C of T in E, let rc ∈ EndT (M ) be the

right multiplication by c. The map cop ’ rc identi¬es C op with the centralizer of

EndE (M ) in EndT (M ).

Proof : Every element f ∈ EndT (M ) in the centralizer of EndE (M ) may be viewed

as an endomorphism of M for its EndE (M )-module structure. By (??), we have

EndEndE (M ) (M ) = E, hence f is right multiplication by some element c ∈ E. Since

f is a T -module endomorphism, c ∈ C.

Suppose now that h : M — M ’ E is a hermitian or skew-hermitian form with

respect to θ. If s : E ’ T is an involution trace, we de¬ne

s— (h) : M — M ’ T

by

s— (h)(x, y) = s h(x, y) for x, y ∈ M .

In view of the properties of s, the form s— (h) is clearly hermitian over T (with

respect to θ) if h is hermitian, and skew-hermitian if h is skew-hermitian. It is

also alternating if h is alternating, since the relation h(x, x) = e ’ θ(e) implies that

s— (h)(x, x) = s(e) ’ θ s(e) .

(4.7) Proposition. If h is nonsingular, then s— (h) is nonsingular and the adjoint

involution σs— (h) on EndT (M ) extends the adjoint involution σh on EndE (M ):

EndE (M ), σh ‚ EndT (M ), σs— (h) .

Moreover, with the notation of (??) and (??),

σs— (h) (rc ) = rθs (c)

for all c ∈ C.

Proof : If x ∈ M is such that s— (h)(x, y) = 0 for all y ∈ M , then h(x, M ) is a right

ideal of E contained in ker s, hence h(x, M ) = {0}. This implies that x = 0 if h is

nonsingular, proving the ¬rst statement.

For f ∈ EndE (M ) and x, y ∈ M we have

h x, f (y) = h σh (f )(x), y .

Hence, applying s to both sides,

s— (h) x, f (y) = s— (h) σh (f )(x), y .

Therefore, σs— (h) (f ) = σh (f ).

On the other hand, for x, y ∈ M and c ∈ C,

s— (h)(xc, y) = s θ(c)h(x, y) .

The de¬ning property of θs shows that the right side is also equal to

s h(x, y)θs (c) = s— (h) x, yθs (c) ,

hence σs— (h) (rc ) = rθs (c) .

(4.8) Example. Suppose that E is central over F , hence the centralizer C of T

in E also is central over F . Let M be a ¬nitely generated right module over E. The

algebra EndE (M ) is a central simple F -subalgebra in EndT (M ), and (??) shows

that its centralizer is isomorphic to C op under the map which carries cop ∈ C op

48 I. INVOLUTIONS AND HERMITIAN FORMS

to the endomorphism rc of right multiplication by c. Hence there is an F -algebra

isomorphism

∼

Ψ : EndE (M ) —F C op ’ EndT (M )

’

which maps f — cop to f —¦ rc = rc —¦ f for f ∈ EndE (M ) and c ∈ C.

Pick an invertible element u ∈ Sym(C, θ) and de¬ne an involution trace s : E ’

T by

s(t — c) = t · TrdC (uc) for t ∈ T , c ∈ C.

The proof of (??) shows that θs = Int(u) —¦ θ. Moreover, (??) shows that for every

nonsingular hermitian or skew-hermitian form h : M —M ’ E, the involution σs— (h)

op

on EndT (M ) corresponds under Ψ to σh —θs where θs (cop ) = θs (c)

op op

for c ∈ C:

∼

Ψ : EndE (M ) —F C op , σh — θs ’ EndT (M ), σs— (h) .

op

’

As a particular case, we may consider T = F , E = C and M = C. Then one

sees that EndE (M ) = C by identifying c ∈ C with left multiplication by c, and the

isomorphism Ψ is the same as in Wedderburn™s theorem (??):

∼

Ψ : C —F C op ’ EndF (C).

’

If h : C — C ’ C is de¬ned by h(x, y) = θ(x)y, then σh = θ and the result above

shows that σTrd— (h) corresponds to θ — θ under Ψ.

(4.9) Example. Suppose that C is the center Z of E, so that

E = T —F Z.

Let N be a ¬nitely generated right module over T and h : N — N ’ T be a

nonsingular hermitian form with respect to θ. Extending scalars to Z, we get a

module NZ = N —F Z over E and a nonsingular hermitian form hZ : NZ —NZ ’ E.

Moreover,

EndE (NZ ) = EndT (N ) —F Z and EndT (NZ ) = EndT (N ) — EndF (Z).

Pick a nonzero linear map : Z ’ F which commutes with θ and let

s = IdT — : E ’ T

be the induced involution trace on E. We claim that under the identi¬cation above,

σs— (hZ ) = σh — σk ,

where k : Z — Z ’ F is the hermitian form de¬ned by

k(z1 , z2 ) = θ(z1 )z2 for z1 , z2 ∈ Z.

Indeed, for x1 , x2 ∈ N and z1 , z2 ∈ Z we have

hZ (x1 — z1 , x2 — z2 ) = h(x1 , x2 ) — θ(z1 )z2 ,

hence

s— (hZ ) = h — k.

We now return to the general case, and show that the involutions on EndT (M )

which are adjoint to transfer forms s— (h) are exactly those which preserve EndE (M )

and induce θs on the centralizer.

§4. HERMITIAN FORMS 49

(4.10) Proposition. Let σ be an involution on EndT (M ) such that

σ EndE (M ) = EndE (M ) σ(re ) = rθs (e)

and

for all e ∈ CE T . There exists a nonsingular hermitian or skew-hermitian form

h : M — M ’ E with respect to θ such that σ = σs— (h) .

Proof : Since θs |Z = θ|Z by (??), it follows that σ(rz ) = rθs (z) = rθ(z) for all

z ∈ Z. Therefore, Theorem (??) shows that the restriction of σ to EndE (M ) is the

adjoint involution with respect to some nonsingular hermitian or skew-hermitian

form h0 : M — M ’ E. Proposition (??) (if θ|F = IdF ) or (??) (if θ|F = IdF )

yields an invertible element u ∈ EndT (M ) such that

σ = Int(u) —¦ σs— (h0 )

and σs— (h0 ) (u) = ±u. By (??), the restriction of σs— (h0 ) to EndE (M ) is σh0 which

is also the restriction of σ to EndE (M ). Therefore, u centralizes EndE (M ). It

follows from (??) that u = re for some e ∈ C — . Proposition (??) shows that

σs— (h0 ) (rc ) = rθs (c) for all c ∈ C, hence

σ(rc ) = u —¦ rθs (c) —¦ u’1 = re’1 θs (c)e .

Since we assume that σ(rc ) = rθs (c) for all c ∈ C, it follows that e ∈ Z — . Moreover,

θ(e) = ±e since σs— (h0 ) (u) = ±u. We may then de¬ne a nonsingular hermitian or

skew-hermitian form h : M — M ’ E by

h(x, y) = e’1 h0 (x, y) for x, y ∈ M .

If δ = θ(e)e’1 (= ±1), we also have

h(x, y) = δh0 (xe’1 , y) = δh0 re’1 (x), y ,

hence

σs— (h) = Int(re ) —¦ σs— (h0 ) = σ.

(4.11) Example. Suppose E is commutative, so that E = Z = C and suppose

that T = F . Assume further that θ = IdE . Let V be a ¬nite dimensional vector

space over F and ¬x some F -algebra embedding

i : Z ’ EndF (V ).

We may then consider V as a vector space over Z by de¬ning

v · z = i(z)(v) for v ∈ V , z ∈ Z.

By de¬nition, the centralizer of i(Z) in EndF (V ) is EndZ (V ).

Suppose that σ is an involution on EndF (V ) which leaves i(Z) elementwise

invariant and that s : Z ’ F is a nonzero linear map. By (??), we have θs = θ =

IdZ . On the other hand, since σ preserves i(Z), it also preserves its centralizer

EndZ (V ). We may therefore apply (??) to conclude that there exists a nonsingular

symmetric or skew-symmetric bilinear form b : V — V ’ Z such that σ = σs— (b) .

By (??), the restriction of σ to EndZ (V ) is σb . If b is symmetric, skew-

symmetric or alternating, then s— (b) has the same property. If char F = 2, the

bilinear form s— (b) cannot be simultaneously symmetric and skew-symmetric, or it

would be singular. Therefore, b and s— (b) are of the same type, and it follows that

σ has the same type as its restriction to EndZ (V ). If char F = 2, it is still true that

50 I. INVOLUTIONS AND HERMITIAN FORMS

σ is symplectic if its restriction to EndZ (V ) is symplectic, since s— (b) is alternating

if b is alternating, but the converse is not true without some further hypotheses.

To construct a speci¬c example, consider a ¬eld Z which is ¬nite dimensional

over its sub¬eld of squares Z 2 , and let F = Z 2 . Pick a nonzero linear map s : Z ’ F

such that s(1) = 0. The nonsingular symmetric bilinear form b on V = Z de¬ned

by b(z1 , z2 ) = z1 z2 is not alternating, but s— (b) is alternating since s vanishes on Z 2 .

Therefore, the involution σs— (b) on EndF (Z) is symplectic, but its restriction σb to

EndZ (Z) is orthogonal. (Indeed, EndZ (Z) = Z and σb = IdZ .)

These observations on the type of an involution compared with the type of its

restriction to a centralizer are generalized in the next proposition.

(4.12) Proposition. Let A be a central simple F -algebra with an involution σ of

the ¬rst kind and let L ‚ A be a sub¬eld containing F . Suppose that σ leaves L

elementwise invariant, so that it restricts to an involution of the ¬rst kind „ on the

centralizer CA L.

(1) If char F = 2, the involutions σ and „ have the same type.

(2) Suppose that char F = 2. If „ is symplectic, then σ is symplectic. If L/F is

separable, then σ and „ have the same type.

Proof : We ¬rst consider the simpler case where char F = 2. If „ is symplectic,

then (??) shows that the centralizer CA L contains an element c such that c+„ (c) =

1. Since „ (c) = σ(c), it also follows from (??) that σ is symplectic.

If „ is orthogonal, then TrdCA L Sym(CA L, „ ) = L by (??). If L/F is separa-

ble, the trace form TL/F is nonzero, hence

TL/F —¦ TrdCA L Sym(CA L, „ ) = F.

Since TL/F —¦ TrdCA L (c) = TrdA (c) for all c ∈ CA L (see Draxl [?, p. 150]), we have

TL/F —¦ TrdCA L Sym(CA L, „ ) ‚ TrdA Sym(A, σ) ,

hence TrdA Sym(A, σ) = {0}, and σ is orthogonal. This completes the proof in

the case where char F = 2.

In arbitrary characteristic, let F be a splitting ¬eld of A in which F is alge-

braically closed and such that the ¬eld extension F /F is separable (for instance, the

function ¬eld of the Severi-Brauer variety SB(A)). The composite L·F (= L—F F )

is then a ¬eld, and it su¬ces to prove the proposition after extending scalars to F .

We may thus assume that A = EndF (V ) for some F -vector space V . If char F = 2

the result then follows from the observations in (??).

(4.13) Corollary. Let M be a maximal sub¬eld of degree n in a central simple

F -algebra A of degree n. Suppose that char F = 2 or that M/F is separable. Every

involution which leaves M elementwise invariant is orthogonal.

Proof : We have CA M = M by (??). Therefore, if σ is an involution on A which

leaves M elementwise invariant, then σ|CA M = IdM and (??) shows that σ is

orthogonal if char F = 2 or M/F is separable.

The result does not hold in characteristic 2 when M/F is not separable, as

example (??) shows.

§4. HERMITIAN FORMS 51

Extension of involutions. The following theorem is a kind of “Skolem-

Noether theorem” for involutions. The ¬rst part is due to M. Kneser [?, p. 37].

(For a di¬erent proof, see Scharlau [?, §8.10].)

(4.14) Theorem. Let B be a simple subalgebra of a central simple algebra A over

a ¬eld F . Suppose that A and B have involutions σ and „ respectively which have

the same restriction to F . Then A has an involution σ whose restriction to B is „ .

If σ is of the ¬rst kind, the types of σ and „ are related as follows:

(1) If char F = 2, then σ can be arbitrarily chosen of orthogonal or symplectic

type, except when the following two conditions hold : „ is of the ¬rst kind and the

degree of the centralizer CA B of B in A is odd. In that case, every extension σ

of „ has the same type as „ .

(2) Suppose that char F = 2. If „ is of symplectic or unitary type, then σ is

symplectic. If „ is of orthogonal type and the center of B is a separable extension

of F , then σ can be arbitrarily chosen of orthogonal or symplectic type, except when

the degree of the centralizer CA B is odd. In that case σ is orthogonal.

Proof : In order to show the existence of σ , we ¬rst reduce to the case where the

centralizer CA B is a division algebra. Let Z = B © CA B be the center of B,

hence also of CA B by the double centralizer theorem (see (??)). Wedderburn™s

theorem (??) yields a decomposition of CA B:

CA B = M · D M —Z D

where M is a matrix algebra: M Mr (Z) for some integer r, and D is a division

algebra with center Z. Let B = B · M B —Z M be the subalgebra of A generated

by B and M . An involution — on Mr (Z) of the same kind as „ can be de¬ned by

letting „ |Z act entrywise and setting

a— = „ (a)t for a ∈ Mr (Z).

The involution „ — — on B — Mr (Z) extends „ and is carried to an involution „

on B through an isomorphism B —Z Mr (Z) B · M = B . It now remains to

extend „ to A. Note that the centralizer of B is a division algebra, i.e., CA B = D.

Since σ and „ have the same restriction to the center F of A, the Skolem-

Noether theorem shows that σ —¦ „ is an inner automorphism. Let σ —¦ „ = Int(u)

for some u ∈ A— , so that

(4.15) σ —¦ „ (x)u = ux for x ∈ B .

Substituting „ (x) for x, we get

σ(x)u = u„ (x) for x ∈ B

and, applying σ to both sides,

(4.16) σ —¦ „ (x)σ(u) = σ(u)x for x ∈ B .

By comparing (??) and (??), we obtain u’1 σ(u) ∈ CA B . At least one of the

elements a+1 = 1 + u’1 σ(u), a’1 = 1 ’ u’1 σ(u) is nonzero, hence invertible since

CA B is a division algebra. If aµ is invertible (where µ = ±1), we have

σ —¦ „ = Int(u) —¦ Int(aµ ) = Int(uaµ )

since aµ ∈ CA B , and uaµ = u + µσ(u) ∈ Sym(A, σ) ∪ Skew(A, σ). Therefore,

σ —¦ Int(uaµ ) (= Int (uaµ )’1 —¦ σ) is an involution on A whose restriction to B is

„ . This completes the proof of the existence of an extension σ of „ to A.

52 I. INVOLUTIONS AND HERMITIAN FORMS

We now discuss the type of σ (assuming that it is of the ¬rst kind, i.e., „ |F =

IdF ). Suppose ¬rst that char F = 2. Since σ extends „ , it preserves B, hence also

its centralizer CA B. It therefore restricts to an involution on CA B. If „ is of the

second kind, we may ¬nd some v ∈ Z — such that σ (v) = ’v. Similarly, if „ is

of the ¬rst kind, then (??) shows that we may ¬nd some v ∈ (CA B)— such that

σ (v) = ’v, except when the degree of CA B is odd. Assuming we have such a v,

the involution σ = Int(v) —¦ σ also extends „ since v ∈ CA B, and it is of the type

opposite to σ since v ∈ Skew(A, σ ) (see (??)). Therefore, „ has extensions of both

types to A.

If the degree of CA B is odd and σ leaves Z elementwise invariant, consider the