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h (x, y) = h v(x), y for x, y ∈ M .
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

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