a nonsingular alternating matrix C ∈ M2g (Q) such that P CP t = 0 and iP CP is

positive de¬nite hermitian. The study of correspondences on the Riemann surface

leads one to consider the matrices M ∈ M2g (Q) for which there exists a matrix

68 I. INVOLUTIONS AND HERMITIAN FORMS

K ∈ Mg (C) such that KP = P M . Following Weyl™s simpler formulation [?],

P

one considers the matrix W = ∈ M2g (C) and the so-called Riemann matrix

P

’iIg 0

R = W ’1 W ∈ M2g (R). The matrices M such that KP = P M for

0 iIg

some K ∈ Mg (C) are exactly those which commute with R. They form a subal-

gebra of M2g (Q) known as the multiplication algebra. As observed by Rosati [?],

this algebra admits the involution X ’ C ’1 X t C (see Weyl [?]). Albert completely

determined the structure of the multiplication algebra in three papers in the An-

nals of Mathematics in 1934“1935. An improved version, [?], see also [?], laid the

foundations of the theory of simple algebras with involutions.

Corollary (??) was observed independently by several authors: see Tits [?,

Proposition 3], Platonov [?, Proposition 5] and Rowen [?, Proposition 5.3].

The original proof of Albert™s theorem on quaternion algebras with involution

of the second kind (??) is given in [?, Theorem 10.21]. This result will be put in a

broader perspective in §??: the subalgebra Q0 is the discriminant algebra of (Q, σ)

(see (??)).

There is an extensive literature on Lie and Jordan structures in rings with

involution; we refer the reader to Herstein™s monographs [?] and [?]. In particu-

lar, Lemma (??) can be proved by ring-theoretic arguments which do not involve

scalar extension (and therefore hold for more general simple rings): see Herstein [?,

Theorem 2.2, p. 28]. In the same spirit, extension of Lie isomorphisms has been

investigated for more general rings: see11 Martindale [?], Rosen [?] and Beidar-

Martindale-Mikhalev [?].

§??. Part (??) of Theorem (??) is due to Albert [?, Theorem 10.19]. Albert

also proved part (??) for crossed products of a special kind: Albert assumes in [?,

Theorem 10.16] the existence of a splitting ¬eld of the form L —F K where L is

Galois over F . Part (??) was stated in full generality by Riehm [?] and proved by

Scharlau [?] (see also [?, §8.9]). In order to see that every central simple algebra

which is Brauer-equivalent to an algebra with involution also has an involution, it

is not essential to use (??): see Albert [?, Theorem 10.12] or Scharlau [?, Corol-

lary 8.8.3]. By combining this result with Exercises ?? and ??, we obtain an

alternate proof of Theorem (??).

§??. If E and E are Brauer-equivalent central simple F -algebras, then Morita

theory yields an E-E -bimodule P and an E -E-bimodule Q such that P —E Q E

and Q —E P E . If M is a right E-module, then there is a natural isomorphism

EndE (M ) = EndE (M —E P ).

Therefore, if E (hence also E ) has an involution, (??) yields one-to-one correspon-

dences between hermitian or skew-hermitian forms on M (up to a central factor),

involutions on EndE (M ) = EndE (M —E P ) and hermitian or skew-hermitian forms

on M —E P (up to a central factor). The correspondence between hermitian forms

can be made more precise and explicit; it is part of a Morita equivalence between

categories of hermitian modules which is discussed in Knus™ book [?, §1.9].

The notion of involution trace was introduced by Fr¨hlich and McEvett [?,

o

§7]. Special cases of the extension theorem (??) have been proved by Rowen [?,

Corollary 5.5] and by Lam-Leep-Tignol [?, Proposition 5.1]. Kneser™s theorem has

11 We are indebted to W. S. Martindale III for references to the recent literature.

NOTES 69

been generalized by Held and Scharlau [?] to the case where the subalgebra is

semisimple. (A particular case of this situation had also been considered by Kneser

in [?, p. 37].)

The existence of involutions for which a given element is symmetric or skew-

symmetric is discussed in Shapiro [?], which also contains an extensive discussion

of the literature.

§??. The de¬nition of quadratic pair in (??) is new. While involutions on

arbitrary central simple algebras have been related to hermitian forms in §??, the

relation between quadratic pairs and quadratic forms is described only in the split

case. The nonsplit case requires an extension of the notion of quadratic form.

For quaternion algebras such an extension was given by Seip-Hornix [?]. Tits [?]

de¬nes a (generalized) quadratic form as an element in the factor group A/ Alt(A, „ )

(compare with (??)); a more geometric viewpoint which also extends this notion

further, was proposed by Bak [?] (see also for instance Hahn-O™Meara [?, 5.1C],

Knus [?, Ch. 1, §5] or Scharlau [?, Ch. 7, §3]).

70 I. INVOLUTIONS AND HERMITIAN FORMS

CHAPTER II

Invariants of Involutions

In this chapter, we de¬ne various kinds of invariants of central simple algebras

with involution (or with quadratic pair) and we investigate their basic properties.

The invariants considered here are analogues of the classical invariants of quadratic

forms: the Witt index, the discriminant, the Cli¬ord algebra and the signature.

How far the analogy can be pushed depends of course on the nature of the in-

volution: an index is de¬ned for arbitrary central simple algebras with involution

or quadratic pair, but the discriminant is de¬ned only for orthogonal involutions

and quadratic pairs, and the Cli¬ord algebra just for quadratic pairs. The Cli¬ord

algebra construction actually splits into two parts: while it is impossible to de¬ne a

full Cli¬ord algebra for quadratic pairs, the even and the odd parts of the Cli¬ord

algebra can be recovered in the form of an algebra and a bimodule. For unitary

involutions, the notion of discriminant turns out to lead to a rich structure: we as-

sociate in §?? a discriminant algebra (with involution) to every unitary involution

on a central simple algebra of even degree. Finally, signatures can be de¬ned for

arbitrary involutions through the associated trace forms. These trace forms also

have relations with the discriminant or discriminant algebra. They yield higher

invariants for algebras with unitary involution of degree 3 in Chapter ?? and for

Jordan algebras in Chapter ??.

The invariants considered in this chapter are produced by various techniques.

The index is derived from a representation of the algebra with involution as the

endomorphism algebra of some hermitian or skew-hermitian space over a division

algebra, while the de¬nitions of discriminant and Cli¬ord algebra are based on

the fact that scalar extension reduces the algebra with quadratic pair to the endo-

morphism algebra of a quadratic space. We show that the discriminant and even

Cli¬ord algebra of the corresponding quadratic form can be de¬ned in terms of

the adjoint quadratic pair, and that the de¬nitions generalize to yield invariants

of arbitrary quadratic pairs. A similar procedure is used to de¬ne the discrimi-

nant algebra of a central simple algebra of even degree with unitary involution.

Throughout most of this chapter, our method of investigation is thus based on

scalar extension: after specifying the de¬nitions “rationally” (i.e., over an arbitrary

base ¬eld), the main properties are proven by extending scalars to a splitting ¬eld.

This method contrasts with Galois descent, where constructions over a separable

closure are shown to be invariant under the action of the absolute Galois group and

are therefore de¬ned over the base ¬eld.

§6. The Index

According to (??), every central simple F -algebra with involution (A, σ) can be

represented as EndD (V ), σh for some division algebra D, some D-vector space V

and some nonsingular hermitian form h on V . Since this representation is essentially

71

72 II. INVARIANTS OF INVOLUTIONS

unique, it is not di¬cult to check that the Witt index w(V, h) of the hermitian space

(V, h), de¬ned as the maximum of the dimensions of totally isotropic subspaces

of V , is an invariant of (A, σ). In this section, we give an alternate de¬nition of this

invariant which does not depend on a representation of (A, σ) as EndD (V ), σh ,

and we characterize the involutions which can be represented as adjoint involutions

with respect to a hyperbolic form. We de¬ne a slightly more general notion of

index which takes into account the Schur index of the algebra. A (weak) analogue

of Springer™s theorem on odd degree extensions is discussed in the ¬nal subsection.

6.A. Isotropic ideals. Let (A, σ) be a central simple algebra with involution

(of any kind) over a ¬eld F of arbitrary characteristic.

(6.1) De¬nition. For every right ideal I in A, the orthogonal ideal I ⊥ is de¬ned

by

I ⊥ = { x ∈ A | σ(x)y = 0 for y ∈ I }.

It is clearly a right ideal of A, which may alternately be de¬ned as the annihilator

of the left ideal σ(I):

I ⊥ = σ(I)0 .

(6.2) Proposition. Suppose the center of A is a ¬eld. For every right ideal I ‚ A,

rdim I + rdim I ⊥ = deg A and I ⊥⊥ = I. Moreover, if (A, σ) = EndD (V ), σh and

I = HomD (V, W ) for some subspace W ‚ V , then

I ⊥ = HomD (V, W ⊥ ).

Proof : Since rdim σ(I) = rdim I, the ¬rst relation follows from the corresponding

statement for annihilators (??). This ¬rst relation implies that rdim I ⊥⊥ = rdim I.

Since the inclusion I ‚ I ⊥⊥ is obvious, we get I = I ⊥⊥ . Finally, suppose I =

HomD (V, W ) for some subspace W ‚ V . For every f ∈ EndD (V ), g ∈ I we have

g(y) ∈ W and h f (x), g(y) = h x, σ(f ) —¦ g(y) for x, y ∈ V .

Therefore, σ(f ) —¦ g = 0 if and only if f (x) ∈ W ⊥ , hence

I ⊥ = HomD (V, W ⊥ ).

A similar result holds if (A, σ) = (E — E op , µ), where E is a central simple F -

algebra and µ is the exchange involution, although the reduced dimension of a right

op

ideal is not de¬ned in this case. Every right ideal I ‚ A has the form I = I1 — I2

where I1 (resp. I2 ) is a right (resp. left) ideal in E, and

op

(I1 — I2 )⊥ = I2 — (I1 )op .

0 0

Therefore, by (??),

dimF I ⊥ = dimF A ’ dimF I and I ⊥⊥ = I

for every right ideal I ‚ A.

In view of the proposition above, the following de¬nitions are natural:

(6.3) De¬nitions. Let (A, σ) be a central simple algebra with involution over a

¬eld F . A right ideal I ‚ A is called isotropic (with respect to the involution

σ) if I ‚ I ⊥ . This inclusion implies rdim I ¤ rdim I ⊥ , hence (??) shows that

rdim I ¤ 1 deg A for every isotropic right ideal.

2

§6. THE INDEX 73

The algebra with involution (A, σ)”or the involution σ itself”is called isotropic

if A contains a nonzero isotropic ideal.

If the center of A is a ¬eld, the index of the algebra with involution (A, σ) is

de¬ned as the set of reduced dimensions of isotropic right ideals:

ind(A, σ) = { rdim I | I ‚ I ⊥ }.

Since the (Schur) index of A divides the reduced dimension of every right ideal,

the index ind(A, σ) is a set of multiples of ind A. More precisely, if (A, σ)

EndD (V ), σh for some hermitian or skew-hermitian space (V, h) over a division

algebra D and w(V, h) denotes the Witt index of (V, h), then ind A = deg D and

ind(A, σ) = { deg D | 0 ¤ ¤ w(V, h) }

since (??) shows that the isotropic ideals of EndD (V ) are of the form HomD (V, W )

with W a totally isotropic subspace of V , and rdim HomD (V, W ) = dimD W deg D.

Thus, if ind(A, σ) contains at least two elements, the di¬erence between two con-

secutive integers in ind(A, σ) is ind A. If ind(A, σ) has only one element, then

ind(A, σ) = {0}, which means that (A, σ) is anisotropic; this is the case for in-

stance when A is a division algebra.

We extend the de¬nition of ind(A, σ) to the case where the center of A is

(E — E op , µ) for some central simple F -algebra E where µ

F — F ; then (A, σ)

is the exchange involution, and we de¬ne ind(A, σ) as the set of multiples of the

1

Schur index of E in the interval [0, 2 deg E]:

deg E

ind(A, σ) = ind E 0¤ ¤ .

2 ind E

(Note that deg E = deg(A, σ) is not necessarily even).

(6.4) Proposition. For every ¬eld extension L/F ,

ind(A, σ) ‚ ind(AL , σL ).

Proof : This is clear if the center of AL = A —F L is a ¬eld, since scalar extensions

preserve the reduced dimension of ideals and since isotropic ideals remain isotropic

under scalar extension. If the center of A is not a ¬eld, the proposition is also clear.

Suppose the center of A is a ¬eld K properly containing F and contained in L;

then by (??),

(A —K L) — (A —K L)op , µ .

(AL , σL )

Since the reduced dimension of every right ideal in A is a multiple of ind A and

since ind(A —K L) divides ind A, the reduced dimension of every isotropic ideal of

(A, σ) is a multiple of ind(A —K L). Moreover, the reduced dimension of isotropic

ideals is bounded by 1 deg A, hence ind(A, σ) ‚ ind(AL , σL ).

2

For central simple algebras with a quadratic pair, we de¬ne isotropic ideals by

a more restrictive condition.

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

a ¬eld F . A right ideal I ‚ A is called isotropic with respect to the quadratic pair

(σ, f ) if the following two conditions hold:

(1) σ(x)y = 0 for all x, y ∈ I.

(2) f (x) = 0 for all x ∈ I © Sym(A, σ).

74 II. INVARIANTS OF INVOLUTIONS

The ¬rst condition means that I ‚ I ⊥ , hence isotropic ideals for the quadratic

pair (σ, f ) are also isotropic for the involution σ. Condition (??) implies that every

x ∈ I ©Sym(A, σ) satis¬es x2 = 0, hence also TrdA (x) = 0. If char F = 2, the map f

1

is the restriction of 2 TrdA to Sym(A, σ), hence condition (??) follows from (??).

Therefore, in this case the isotropic ideals for (σ, f ) are exactly the isotropic ideals

for σ.

The algebra with quadratic pair (A, σ, f )”or the quadratic pair (σ, f ) itself”is

called isotropic if A contains a nonzero isotropic ideal for the quadratic pair (σ, f ).

(6.6) Example. Let (V, q) be an even-dimensional quadratic space over a ¬eld F

of characteristic 2, and let (σq , fq ) be the corresponding quadratic pair on EndF (V )

(see (??)). A subspace W ‚ V is totally isotropic for q if and only if the right ideal

HomF (V, W ) ‚ EndF (V ) is isotropic for (σq , fq ).

Indeed, the standard identi¬cation •q (see (??)) identi¬es HomF (V, W ) with

W —F V . The elements in W — V which are invariant under the switch involution

(which corresponds to σq under •q ) are spanned by elements of the form w —w with

w ∈ W , and (??) shows that fq —¦ •q (w — w) = q(w). Therefore, the subspace W

is totally isotropic with respect to q if and only if fq vanishes on HomF (V, W ) ©

Sym EndF (V ), σq . Proposition (??) shows that this condition also implies that

HomF (V, W ) is isotropic with respect to the involution σq .

Mimicking (??), we de¬ne the index of a central simple algebra with quadratic

pair (A, σ, f ) as the set of reduced dimensions of isotropic ideals:

ind(A, σ, f ) = { rdim I | I is isotropic with respect to (σ, f ) }.

6.B. Hyperbolic involutions. Let E be a central simple algebra with invo-

lution θ (of any kind) and let U be a ¬nitely generated right E-module. As in (??),

we use the involution θ to endow the dual of U with a structure of right E-module

θ—

U . For » = ±1, de¬ne

h» : ( θ U — • U ) — ( θ U — • U ) ’ E

by

h» (θ • + x, θ ψ + y) = •(y) + »θ ψ(x)

for •, ψ ∈ U — and x, y ∈ U . Straightforward computations show that h1 (resp.

h’1 ) is a nonsingular hermitian (resp. alternating) form on θ U — • U with respect

to the involution θ on E. The hermitian or alternating module (θ U — • U, h» ) is

denoted H» (U ). A hermitian or alternating module (M, h) over (E, θ) is called

hyperbolic if it is isometric to some H» (U ).

The following proposition characterizes the adjoint involutions with respect to

hyperbolic forms:

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

kind ) over a ¬eld F of arbitrary characteristic. Suppose the center K of A is a

¬eld. The following conditions are equivalent:

(1) for every central simple K-algebra E Brauer-equivalent to A and every involu-

tion θ on E such that θ|K = σ|K , every hermitian or skew-hermitian module (M, h)

over (E, θ) such that (A, σ) EndE (M ), σh is hyperbolic;

(2) there exists a central simple K-algebra E Brauer-equivalent to A, an involution

θ on E such that θ|K = σ|K and a hyperbolic hermitian or skew-hermitian module

(M, h) over (E, θ) such that (A, σ) EndE (M ), σh ;

§6. THE INDEX 75

(3) 1 deg A ∈ ind(A, σ) and further, if char F = 2, the involution σ is either sym-

2

plectic or unitary;

(4) there is an idempotent e ∈ A such that σ(e) = 1 ’ e.

Proof : (??) ’ (??) This is clear.

(??) ’ (??) In the hyperbolic module M = θ U — • U , the submodule U is

totally isotropic, hence the same argument as in (??) shows that the right ideal

HomE (M, U ) is isotropic in EndE (M ). Moreover, since rdim U = 1 rdim M , we

2

have rdim HomE (M, U ) = 2 deg EndE (M ). Therefore, (??) implies that 1 deg A ∈

1

2

ind(A, σ). If char F = 2, the hermitian form h+1 = h’1 is alternating, hence σ is

symplectic if the involution θ on E is of the ¬rst kind, and is unitary if θ is of the

second kind.

1

(??) ’ (??) Let I ‚ A be an isotropic ideal of reduced dimension 2 deg A.

By (??), there is an idempotent f ∈ A such that I = f A. Since I is isotropic, we

have σ(f )f = 0. We shall modify f into an idempotent e such that I = eA and

σ(e) = 1 ’ e.

The ¬rst step is to ¬nd u ∈ A such that σ(u) = 1 ’ u. If char F = 2, we

may choose u = 1/2; if char F = 2 and σ is symplectic, the existence of u follows

from (??); if char F = 2 and σ is unitary, we may choose u in the center K of A,

since K/F is a separable quadratic extension and the restriction of σ to K is the

nontrivial automorphism of K/F .

We next set e = f ’f uσ(f ) and proceed to show that this element satis¬es (??).

Since f 2 = f and σ(f )f = 0, it is clear that e is an idempotent and σ(e)e = 0.

Moreover, since σ(u) + u = 1,

eσ(e) = f σ(f ) ’ f uσ(f ) ’ f σ(u)σ(f ) = 0.

Therefore, e and σ(e) are orthogonal idempotents; it follows that e + σ(e) also is an

idempotent, and e + σ(e) A = eA • σ(e)A. To complete the proof of (??), observe

that e ∈ f A and f = ef ∈ eA, hence eA = f A = I. Since rdim I = 1 deg A, 2

1

it follows that dimF eA = dimF σ(e)A = 2 dimF A, hence e + σ(e) A = A and

therefore e + σ(e) = 1.

(??) ’ (??) Let E be a central simple K-algebra Brauer-equivalent to A and

θ be an involution on E such that θ|K = σ|K . Let also (M, h) be a hermitian or

skew-hermitian module over (E, θ) such that (A, σ) EndE (M ), σh . Viewing

this isomorphism as an identi¬cation, we may ¬nd for every idempotent e ∈ A a

pair of complementary submodules U = im e, W = ker e in M such that e is the

projection M ’ U parallel to W ; then 1 ’ e is the projection M ’ W parallel

to U and σ(e) is the projection M ’ W ⊥ parallel to U ⊥ . Therefore, if σ(e) = 1 ’ e

∼

we have U = U ⊥ and W = W ⊥ . We then de¬ne an isomorphism W ’ θ U — by ’

mapping w ∈ W to θ •w where •w ∈ U — is de¬ned by •w (u) = h(w, u) for u ∈ U .

∼

This isomorphism extends to an isometry M = W • U ’ H» (U ) where » = +1 if

’

h is hermitian and » = ’1 if h is skew-hermitian.

(6.8) De¬nition. A central simple algebra with involution (A, σ) over a ¬eld F ”

or the involution σ itself”is called hyperbolic if either the center of A is isomorphic

to F — F or the equivalent conditions of (??) hold. If the center is F — F , then

the idempotent e = (1, 0) satis¬es σ(e) = 1 ’ e; therefore, in all cases (A, σ) is

hyperbolic if and only if A contains an idempotent e such that σ(e) = 1 ’ e. This

condition is also equivalent to the existence of an isotropic ideal I of dimension

1

dimF I = 2 dimF A if char F = 2, but if char F = 2 the extra assumption that σ is

76 II. INVARIANTS OF INVOLUTIONS

symplectic or unitary is also needed. For instance, if A = M2 (F ) (with char F = 2)

1

and σ is the transpose involution, then 2 deg A ∈ ind(A, σ) since the right ideal

I = { x x | x, y ∈ F } is isotropic, but (A, σ) is not hyperbolic since σ is of

yy

orthogonal type.

Note that (A, σ) may be hyperbolic without A being split; indeed we may have

1 1

ind(A, σ) = {0, 2 deg A}, in which case the index of A is 2 deg A.

From any of the equivalent characterizations in (??), it is clear that hyper-

bolic involutions remain hyperbolic over arbitrary scalar extensions. Characteriza-

tion (??) readily shows that hyperbolic involutions are also preserved by transfer.

Explicitly, consider the situation of §??: 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, so

that E = T —F C where C is the centralizer of T in E. Let θ be an involution on E

which preserves T and let s : E ’ T be an involution trace. Recall from (??) that

for every hermitian or skew-hermitian module (M, h) over (E, θ) there is a transfer

M, s— (h) which is a hermitian or skew-hermitian module over (T, θ).

(6.9) Proposition. If h is hyperbolic, then s— (h) is hyperbolic.

Proof : If h is hyperbolic, (??) yields an idempotent e ∈ EndE (M ) such that

σh (e) = 1 ’ e. By (??), the involution σs— (h) on EndT (M ) extends σh , hence

e also is an idempotent of EndT (M ) such that σs— (h) (e) = 1 ’ e. Therefore, s— (h)

is hyperbolic.

In the same spirit, we have the following transfer-type result:

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

kind ) over a ¬eld F and let L/F be a ¬nite extension of ¬elds. 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. If (AL , σL ) is hyperbolic,

then A —F EndF (L), σ — ν is hyperbolic.

Proof : The embedding L ’ EndF (L) induces an embedding

(AL , σL ) = (A —F L, σ — IdL ) ’ A —F EndF (L), σ — ν .

The same argument as in the proof of (??) applies. (Indeed, (??) may be regarded

as the special case of (??) where C = Z = L: see ??).

(6.11) Example. Interesting examples of hyperbolic involutions can be obtained

as follows: 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 u ∈ Sym(A, σ)©A— be a symmetric

unit in A. De¬ne an involution νu on M2 (A) by

’σ(a21 )u’1

a11 a12 σ(a11 )

νu =

’uσ(a12 ) uσ(a22 )u’1

a21 a22

for a11 , a12 , a21 , a22 ∈ A, i.e.,

10

νu = Int —¦ (σ — t)

0 ’u

where t is the transpose involution on M2 (F ).

Claim. The involution νu is hyperbolic if and only if u = vσ(v) for some v ∈ A.

§6. THE INDEX 77

Proof : Let D be a division algebra Brauer-equivalent to A and let θ be an involution

on D of the same type as σ. We may identify (A, σ) = EndD (V ), σh for some

hermitian space (V, h) over (D, σ), by (??). De¬ne a hermitian form h on V by

h (x, y) = h u’1 (x), y = h x, u’1 (y) for x, y ∈ V .

Under the natural identi¬cation M2 (A) = EndD (V • V ), the involution νu is the

adjoint involution with respect to h⊥(’h ). The form h⊥(’h ) is hyperbolic if and

only if (V, h) is isometric to (V, h ). Therefore, by Proposition (??), νu is hyperbolic

if and only if (V, h) is isometric to (V, h ). This condition is also equivalent to the

existence of a ∈ A— such that h (x, y) = h a(x), a(y) for all x, y ∈ V ; in view of

the de¬nition of h , this relation holds if and only if u = a’1 σ(a’1 ).

The fact that νu is hyperbolic if u = vσ(v) for some v ∈ A can also be readily

1 1 ’1

proved by observing that the matrix e = 2 v v 1 is an idempotent such that

νu (e) = 1 ’ e.

Hyperbolic quadratic pairs. By mimicking characterization (??) of hyper-

bolic involutions, we may de¬ne hyperbolic quadratic pairs as follows:

(6.12) De¬nition. A quadratic pair (σ, f ) on a central simple algebra A of even

1

degree over a ¬eld F of arbitrary characteristic is called hyperbolic if 2 deg A ∈