Hint: Re¬ne the argument used in the proof of Proposition (??).

6. Let σ, „ be distinct symplectic involutions on a biquaternion F -algebra A. Show

that dimF Symd(A, σ)©Symd(A, „ ) = 2. Show also that there is a quaternion

algebra B ‚ A over some quadratic extension of F such that σ|B = „ |B is the

conjugation involution, and that the algebra B is uniquely determined by this

condition.

7. Let σ, „ , θ be symplectic involutions on a biquaternion F -algebra A. Show that

the invariants of these involutions are related by jσ („ ) = j„ (σ) and jσ („ ) +

j„ (θ) + jθ (σ) ∈ I 3 Wq F . Use this result to show that if σ and „ are conjugate,

then jσ (θ) = j„ (θ).

8. Let σ be a symplectic involution on a biquaternion F -algebra A. Let

Symd(A, σ)0 = { x ∈ Symd(A, σ) | Trpσ (x) = 0 }

and let sσ : Symd(A, σ)0 ’ F be the squaring map. Show that ind(A) ¤ 2 if

and only if sσ is a subform of some (uniquely determined) 3-fold P¬ster form

πσ . Suppose these conditions hold; then

(a) show that (A, σ) has a decomposition

(A, σ) = M2 (F ), σ1 —F (Q, γ)

for some quaternion algebra Q with canonical involution γ and some or-

thogonal involution σ1 on M2 (F ), and that πσ = disc σ1 · NrdQ ;

(b) for θ a hyperbolic involution on A, show that πσ = jθ (σ);

(c) show that G(A, σ) = G(πσ ) = Sn(sσ ).

Hint: The equality G(A, σ) = Sn(sσ ) follows from (??) and G(A, σ) =

G(πσ ) follows from (??) and (??).

9. Suppose char F = 2. Let K/F be an ´tale quadratic extension with non-

e

—

trivial automorphism ι and let δ ∈ K be such that ι(δ) = ’δ. Let (V, q)

be an odd-dimensional quadratic space over K with trivial discriminant and

let ζ ∈ C(V, q) be an orientation of (V, q). De¬ne an F -linear map i : V ’

M2 NK/F C0 (V, q) by

’δ ι (x · ζ) — 1 ’ 1 — (x · ζ)

0

i(x) = ι

(x · ζ) — 1 + 1 — (x · ζ) 0

for x ∈ V . Show that the map i induces an F -algebra isomorphism:

∼

i— : C V, (trK/F )— ( δ · q) ’ M2 NK/F C0 (V, q) .

’

Use this result to give a direct proof of the fact that if Q is a quaternion K-

algebra and s : Q0 ’ K is the squaring map on the space of pure quaternions,

then Q0 , (trK/F )— ( δ · s) is an Albert quadratic space of NK/F (Q).

10. Suppose char F = 2. Let Q1 , Q2 be quaternion F -algebras with canonical in-

volutions γ1 , γ2 and let (A, θ) = (Q1 , γ1 ) —F (Q2 , γ2 ). De¬ne a linear endomor-

phism p on Skew(A, θ) = (Q0 —1)•(1—Q0) by p(x1 —1+1—x2) = x1 —1’1—x2

1 2

and a quadratic form q : Skew(A, θ) ’ F by q(x) = xp(x). Consider another

pair of quaternion F -algebras Q1 , Q2 and (A , θ ) = (Q1 , γ1 ) —F (Q2 , γ2 ), and

de¬ne p , q on Skew(A , θ ) as p, q were de¬ned on Skew(A, θ). Show that

∼

for every isomorphism f : (A, θ) ’ (A , θ ) there exists some » ∈ F — such

’

272 IV. ALGEBRAS OF DEGREE FOUR

∼

that f ’1 —¦ p —¦ f = »p, hence f restricts to a similitude (Skew(A, θ), q) ’ ’

Skew(A , θ ), q with multiplier ».

Hint: Use (??).

This exercise is inspired by Knus-Parimala-Sridharan [?, Theorem 3.4] and

Wadsworth [?]. It shows that the forms q and q are similar without using the

fact that they are Albert forms of A and A .

11. Let σ be a symplectic involution on a biquaternion algebra A. Show that

the invariant jσ of symplectic involutions and the homomorphism ±σ : “ ’

I 3 Wq F/I 4 Wq F are related by

±σ Nrpσ (v), v = jσ Int(v) —¦ σ ’ Nrpσ (v) · Nrpσ +I 4 Wq F

for all v ∈ Symd(A, σ)— .

12. Suppose char F = 2. Let A be a biquaternion division algebra and let x ∈

SL1 (A). √

(a) Let σ be an arbitrary symplectic involution and let L = F ( a) be a

quadratic extension of F in Sym(A, σ) which contains σ(x)x. Recall from

(??) that one can ¬nd b, c, d ∈ F — such that A (a, b)F — (c, d)F . Show

that L contains an element y such that σ(x)x = yy ’1 where is the

nontrivial automorphism of L/F , and that

±(x) = NL/F (y), b, c, d + I 5 F.

(b) Suppose x is contained in a maximal sub¬eld E ‚ A which contains an

√

intermediate quadratic extension L = F ( a). Recall from (??) that A

(a, b)F — (c, d)F for some b, c, d ∈ F — . Show that L contains an element y

such that NE/L (x) = yy ’1 where is the nontrivial automorphism of L/F ,

and that

±(x) = NL/F (y), b, c, d + I 5 F.

13. Let Q be a quaternion F -algebra. Show that every element in SL1 (Q) is a

commutator, except if F = F2 (the ¬eld with two elements).

Hint: Argue as in (??). If q ∈ SL1 (Q) and 1 + q is not invertible and

∼

nonzero, then show that there is an isomorphism Q ’ M2 (F ) which identi¬es

’

q with ’1 ’1 . Then q = xyx’1 y ’1 with x = 1 ’1 and y = ’1/2 1 if

1 0

0

0 ’1 0

»+1 0 1 » if char F = 2 and » = 0, 1.

char F = 2, with x = 0 » and y = 0 1

14. Let Q be a quaternion division F -algebra and let (V, h) be a nonsingular her-

mitian space of dimension 2 over Q with respect to the canonical involution,

so that EndQ (V ), σh is a central simple algebra of degree 4 with symplectic

involution. Let g ∈ Sp(V, h) be such that IdV + g is not invertible, and let

v1 ∈ V be a nonzero vector such that g(v1 ) = ’v1 .

(a) If v1 is anisotropic, show that for each vector v2 ∈ V which is orthogonal

to v1 , there is some ± ∈ Q such that g(v2 ) = v2 ± and NrdQ (±) = 1.

(b) If v1 is isotropic, show that for each isotropic vector v2 ∈ V such that

h(v1 , v2 ) = 1, there is some β ∈ Q such that g(v2 ) = v1 β ’ v2 and

TrdQ (β) = 0.

15. Let b be a nonsingular alternating bilinear form on a 4-dimensional F -vector

space V and let g ∈ Sp(V, b) be such that IdV + g is not invertible.

(a) Show that ker(IdV + g) = im(IdV + g)⊥ .

(b) If the rank of IdV + g is 1 or 2, show that V = V1 • V2 for some 2-

dimensional subspaces V1 , V2 which are preserved under g, hence g can be

EXERCISES 273

represented by the matrix

« «

’1 1 0 0 ’1 1 0 0

¬ 0 ’1 0 0· ¬ 0 ’1 0 0·

g1 = ¬ · or g2 = ¬ ·.

0 0 ’1 0 0 0 ’1 1

0 0 0 ’1 0 0 0 ’1

Use Exercise ?? to conclude that g is a commutator if F = F2 . If char F =

2, show that g1 = a1 b1 a’1 b’1 and g2 = a2 b2 a’1 b’1 , where

1 1 2 2

« «

1010 1000

¬0 1 0 0 · ¬ ·

· , b1 = ¬0 1 0 0· ,

a1 = ¬0 0 1 0 0 1 1 0

0001 0001

« «

0010 1000

¬0 0 0 1· ¬ ·

· , b2 = ¬0 1 0 0· .

a2 = ¬

1 0 0 0 0 0 1 1

0100 0001

(c) If IdV + g has rank 3, show that g can be represented by the matrix

«

’1 1 0 0

¬ 0 ’1 1 0·

¬ ·

0 1

0 ’1

0 0 0 ’1

(with respect to a suitable basis).

If char F = 2, let

«

’3 ’8 22 ’12

¬’3 ’9 23 ’12·

y=¬ ·.

’3 ’9 22 ’11

’4 ’5 16 ’8

Show that (x’1)3 (x+1) is the minimum and the characteristic polynomial

of both matrices gy and y. Conclude that there is an invertible matrix x

such that gy = xyx’1 , hence g is a commutator. If char F = 2, let

«

0100

¬1 0 0 1·

z=¬ ·

1 0 1 1 .

1110

Show that x4 + x3 + x2 + x + 1 is the minimum and the characteristic

polynomial of both matrices gz and z. As in the preceding case, conclude

that g is a commutator.

16. Let A be a biquaternion F -algebra with symplectic involution σ. Write simply

V = Symd(A, σ), V 0 = Symd(A, σ)0 and q = Nrpσ , and let q 0 be the restriction

of q to V 0 . Using (??), show that there is a canonical isomorphism SL1 (A)

Spin(V, q) which maps the subgroup [A— , A— ] to the subgroup

Spin (V, q) = Spin(V 0 , q 0 ) · [„¦, “+ (V 0 , q 0 )]

where „¦ = „¦ EndF (V ), σq , fq is the extended Cli¬ord group. Deduce that

the subgroup Spin (V, q) ‚ Spin(V, q) is generated by the subgroups Spin(U )

for all the proper nonsingular subspaces U ‚ V .

274 IV. ALGEBRAS OF DEGREE FOUR

Hint: Use the proof of (??).

17. Suppose char F = 2 and F contains a primitive 4th root of unity ζ. Let A =

(a, b)F — (c, d)F be a biquaternion division F -algebra.

(a) Show that if A is cyclic (i.e., if A contains a maximal sub¬eld which is

cyclic over F ), then a, b, c, d is hyperbolic.

(b) (Morandi-Sethuraman [?, Proposition 7.3]) Suppose d is an indeterminate

over some sub¬eld F0 containing a, b, c, and F = F0 (d). Show that A is

cyclic if and only if a, b, c is hyperbolic.

Hint: If a, b, c is hyperbolic, the following equation has a nontrivial

solution in F0 :

a(x2 + cx2 ) + b(y1 + cy2 ) ’ ab(z1 + cz2 ) = 0.

2 2 2 2

1 2

√ √ √√

Let e = 2 c(ax1 x2 + by1 y2 ’ abz1 z2 ) ∈ F0 ( c). Show that F0 ( c)( e) is

cyclic over F0 and splits (a, b)F (√c) , hence also A.

Notes

§??. If char F = 2, an alternative way to de¬ne the canonical isomorphism

N —¦ C(A, σ) (A, σ) for (A, σ) ∈ D2 (in (??)) is to re¬ne the fundamental re-

lation (??) by taking the involutions into account. As pointed out in the notes

for Chapter ??, one can de¬ne a nonsingular hermitian form H on the left A-

submodule B (A, σ) ‚ B(A, σ) of invariant elements under the canonical involution

ω and show that the canonical isomorphism ν of (??) restricts to an isomorphism

of algebras with involution:

∼

NZ/F C(A, σ), σ ’ EndA B (A, σ), σH

’

where Z is the center of the Cli¬ord algebra C(A, σ). Since deg A = 4, dimension

count shows that the canonical map b : A ’ B(A, σ) induces an isomorphism of A-

∼

modules A ’ B (A, σ). Moreover, under this isomorphism, H (a1 , a2 ) = 2a1 σ(a2 )

’

for a1 , a2 ∈ A. Therefore, b induces a canonical isomorphism

∼

(A, σ) ’ EndA B (A, σ), σH .

’

Similarly, in Theorem (??) the canonical isomorphism D —¦ C(A, σ) (A, σ) for

(A, σ) ∈ D3 can be derived from properties of the bimodule B(A, σ). De¬ne a left

C(A, σ) —Z C(A, σ)-module structure on B(A, σ) by

(c1 — c2 ) u = c1 — u · σ(c2 ) for c1 , c2 ∈ C(A, σ), u ∈ B(A, σ).

If g ∈ C(A, σ) —Z C(A, σ) is the Goldman element of the Cli¬ord algebra, the map

C(A, σ) —Z C(A, σ) ’ B(A, σ) which carries ξ to ξ 1b induces an isomorphism of

left C(A, σ) —Z C(A, σ)-modules

∼

[C(A, σ) —Z C(A, σ)](1 ’ g) ’ B(A, σ).

’

We thus get a canonical isomorphism

»2 C(A, σ) EndC(A,σ)—Z C(A,σ) B(A, σ).

Using (??), we may identify the right-hand side with A —F Z and use this identi¬-

cation to get a canonical isomorphism D C(A, σ), σ A.

In the proofs of Theorems (??) and (??), it is not really necessary to consider

the split cases separately if char F = 0; one can instead use results on the extension

NOTES 275

of Lie algebra isomorphisms in Jacobson [?, Ch. 10, §4] to see directly that the

™1 1 ™™ 1 ™

1

canonical Lie algebra isomorphisms n—¦ 2 c, 2 c—¦n, »2 —¦ 2 c and 2 c—¦»2 extend (uniquely)

to isomorphisms of the corresponding algebras with involution (see Remarks (??)

and (??)).

The fact that a central simple algebra of degree 4 with orthogonal involution

decomposes into a tensor product of stable quaternion subalgebras if and only if

the discriminant of the involution is trivial (Corollary (??) and Proposition (??))

was proved by Knus-Parimala-Sridharan [?, Theorem 5.2], [?]. (The paper [?] deals

with Azumaya algebras over rings in which 2 is invertible, whereas [?] focuses on

central simple algebras, including the characteristic 2 case).

The description of groups of similitudes of quadratic spaces of dimension 3, 4,

5 and 6 dates back to Van der Waerden [?] and Dieudonn´ [?, §3]. The case of

e

quadratic spaces over rings was treated by Knus [?, §3] and by Knus-Parimala-

Sridharan [?, §6] (see also Knus [?, Ch. 5]). Cli¬ord groups of algebras of degree 4

or 6 with orthogonal involution are determined in Merkurjev-Tignol [?, 1.4.2, 1.4.3].

§??. Albert forms are introduced in Albert [?]. Theorem 3 of that paper yields

the criterion for the biquaternion algebra A = (a, b)F — (c, d)F to be a division alge-

bra in terms of the associated quadratic form q = a, b, ’ab, ’c, ’d, cd . (See (??).)

The de¬nition of the form q thus depends on a particular decomposition of the bi-

quaternion algebra A. The fact that quadratic forms associated to di¬erent decom-

positions are similar was ¬rst proved by Jacobson [?, Theorem 3.12] using Jordan

structures, and later by Knus [?, Proposition 1.14] and Mammone-Shapiro [?] us-

ing the algebraic theory of quadratic forms. (See also Knus-Parimala-Sridharan [?,

Theorem 4.2].) Other proofs of Albert™s Theorem (??) were given by P¬ster [?,

p. 124] and Tamagawa [?] (see also Seligman [?, App. C]). A notion of Albert form

in characteristic 2 is discussed in Mammone-Shapiro [?]. Note that the original

version of Jacobson™s paper [?] does not cover the characteristic 2 case adequately;

see the reprinted version in Jacobson™s Collected Mathematical Papers [?], where

the characteristic is assumed to be di¬erent from 2.

From the de¬nition of the quadratic form q = a, b, ’ab, ’c, ’d, cd associated

to A = (a, b)F — (c, d)F , it is clear that q is isotropic if and only if the quaternion

algebras (a, b)F and (c, d)F have a common maximal sub¬eld. Thus Corollary (??)

easily follows from Albert™s Theorem (??). The proof given by Albert in [?] is more

direct and also holds in characteristic 2. Another proof (also valid in characteris-

tic 2) was given by Sah [?]. If the characteristic is 2, the result can be made more

precise: if a tensor product of two quaternion division algebras is not a division

algebra, then the two quaternion algebras have a common maximal sub¬eld which

is a separable extension of the center. This was ¬rst shown by Draxl [?]. The proofs

given by Knus [?] and by Tits [?] work in all characteristics, and yield the more

precise result in characteristic 2.

The fact that G(q) = F —2 · NrdA (A— ) for an Albert form q of a biquaternion

algebra A is already implicit in Van der Waerden [?, pp. 21“22] and Dieudonn´ [?, e

Nos 28, 30, 34]. Knus-Lam-Shapiro-Tignol [?] gives other characterizations of that

group. In particular, it is shown that this group is also the set of discriminants of

orthogonal involutions on A; see Parimala-Sridharan-Suresh [?] for another proof

of that result.

If σ is a symplectic involution on a biquaternion algebra A, the (Albert) quad-

ratic form Nrpσ de¬ned on the space Symd(A, σ) is the generic norm of Symd(A, σ),

viewed as a Jordan algebra, see Jacobson [?]. This is the point of view from which

276 IV. ALGEBRAS OF DEGREE FOUR

results on Albert forms are derived in Jacobson [?, Ch. 6, §4]. The invariant jσ („ )

of symplectic involutions is de¬ned in a slightly di¬erent fashion in Knus-Lam-

Shapiro-Tignol [?, §3]. See (??) for the relation between jσ („ ) and Rost™s higher

cohomological invariants.

If σ is an orthogonal involution on a biquaternion algebra A, the linear endomor-

phism pσ of Skew(A, σ) was ¬rst de¬ned by Knus-Parimala-Sridharan [?], [?] (see

also Knus [?, Ch. 5]), who called it the pfa¬an adjoint because of its relation with

the pfa¬an. Pfa¬an adjoints for algebras of degree greater than 4 are considered

in Knus-Parimala-Sridharan [?]. If deg A = 2m, the pfa¬an adjoint is a polynomial

map of degree m ’ 1 from Skew(A, σ) to itself. Knus-Parimala-Sridharan actually

treat orthogonal and symplectic involutions simultaneously (and in the context of

Azumaya algebras): if σ is a symplectic involution on a biquaternion algebra A,

the pfa¬an adjoint is the endomorphism of Sym(A, σ) de¬ned in §??.

Further results on Albert forms can be found in Lam-Leep-Tignol [?], where

maximal sub¬elds of a biquaternion algebra are investigated; in particular, neces-

sary and su¬cient conditions for the cyclicity of a biquaternion algebra are given

in that paper.

§??. Suppose char F = 2. As observed in the proof of (??), the group “ =

{ (», a) ∈ F — —A— | »2 = NrdA (a) }, for A a biquaternion F -algebra, can be viewed

as the special Cli¬ord group of the quadratic space Sym(A, σ), Nrpσ for any

symplectic involution σ. The map ±σ : “ ’ I 4 F/I 5 F can actually be de¬ned on the

full Cli¬ord group “ Sym(A, σ), Nrpσ by mapping the generators v ∈ Sym(A, σ)—

to ¦v + I 5 F . Showing that this map is well-de¬ned is the main di¬culty of this

approach.

The homomorphism ± : SK1 (A) ’ I 4 F/I 5 F (for A a biquaternion algebra) was

originally de¬ned by Rost in terms of Galois cohomology, as a map ± : SK1 (A) ’

H 4 (F, µ2 ). Rost also proved exactness of the following sequence:

± h

0 ’ SK1 (A) ’ H 4 (F, µ2 ) ’ H 4 (F (q), µ2 )

’ ’

where h is induced by scalar extension to the function ¬eld of an Albert form

q, see Merkurjev [?, Theorem 4] and Kahn-Rost-Sujatha [?]. The point of view

of quadratic forms developed in §?? is equivalent, in view of the isomorphism

∼

e4 : I 4 F/I 5 F ’ H 4 (F, µ2 ) proved by Rost (unpublished) and more recently by

’

Voevodsky, which leads to a commutative diagram

± i

0 ’ ’ ’ SK1 (A) ’ ’ ’ I 4 F/I 5 F ’ ’ ’ I 4 F (q)/I 5 F (q)

’’ ’’ ’’

¦ ¦

¦ ¦

e e

4 4

± h

0 ’ ’ ’ SK1 (A) ’ ’ ’ H 4 (F, µ2 ) ’ ’ ’ H 4 F (q), µ2 .

’’ ’’ ’’

The fact that the sequences above are zero-sequences readily follows from functo-

riality of ± and ± , since SK1 (A) = 0 if A is not a division algebra or, equivalently

by (??), if q is isotropic. In order to derive exactness of the sequence above from

exactness of the sequence below, only “elementary” information on the map e 4 is

needed: it is su¬cient to use the fact that on P¬ster forms e4 is well-de¬ned (see

Elman-Lam [?, 3.2]) and injective (see Arason-Elman-Jacob [?, Theorem 1]). In

fact, (??) shows that the image of ± consists of 4-fold P¬ster forms (modulo I 5 F )

and, on the other hand, the following proposition also shows that every element in

ker i is represented by a single 4-fold P¬ster form:

NOTES 277

(17.30) Proposition. If q is an Albert form which represents 1, then

ker i = { π + I 5 F | π is a 4-fold P¬ster form containing q }.

Proof : By Fitzgerald [?, Corollary 2.3], the kernel of the scalar extension map

W F ’ W F (q) is an ideal generated by 4-fold and 5-fold P¬ster forms. Therefore,

every element in ker i is represented by a sum of 4-fold P¬ster forms which become

hyperbolic over F (q). By the Cassels-P¬ster subform theorem (see Scharlau [?,

Theorem 4.5.4]) the P¬ster forms which satisfy this condition contain a subform

isometric to q. If π1 , π2 are two such 4-fold P¬ster forms, then the Witt index of

π1 ⊥ ’π2 is at least dim q = 6. By Elman-Lam [?, Theorem 4.5] it follows that

and elements a1 , a2 ∈ F — such that

there exists a 3-fold P¬ster form

π1 = · a1 and π2 = · a2 .

Therefore,

mod I 5 F.

π1 + π 2 ≡ · a 1 a2

By induction on the number of terms, it follows that every sum of 4-fold P¬ster

forms representing an element of ker i is equivalent modulo I 5 F to a single 4-fold

P¬ster form.

The image of the map ±σ can be described in a similar way. Since ±σ = 0 if σ

is hyperbolic or, equivalently by (??), if the 5-dimensional form sσ is isotropic, it

follows by functoriality of ±σ that im ±σ lies in the kernel of the scalar extension

map to F (sσ ). One can use the arguments in the proof of Merkurjev [?, Theorem 4]

to show that this inclusion is an equality, so that the following sequence is exact:

±

“σ ’’ I 4 F/I 5 F ’’ I 4 F (sσ )/I 5 F (sσ ).

σ

’

Corollary (??) shows that every element in [A— , A— ] is a product of two com-

mutators. If A is split, every element in [A— , A— ] can actually be written as a single

commutator, as was shown by Thompson [?]. On the other hand, Kursov [?] has

found an example of a biquaternion algebra A such that the group [A— , A— ] does

not consist of commutators, hence our lower bound for the number of factors is

sharp, in general.

The ¬rst example of a biquaternion algebra A such that SK1 (A) = 0 is due

to Platonov [?]. For a slightly di¬erent relation between the reduced Whitehead

group of division algebras and Galois cohomology of degree 4, see Suslin [?].

Along with the group Σσ (A), the group generated by skew-symmetric units in a

central simple algebra with involution (A, σ) is also discussed in Yanchevski˜ [?]. If ±

—

σ is orthogonal, it is not known whether this subgroup is normal in A . Triviality of

the group USK1 (B, „ ) can be shown not only for division algebras of prime degree,

but also for division algebras whose degree is square-free; indeed, the exponent of

USK1 (B, „ ) divides deg B/p1 . . . pr , where p1 , . . . , pr are the prime factors of deg B:

see Yanchevski˜ [?]. Examples where the group K1 Spin(A) is not trivial are given

±

in Monastyrny˜ ±-Yanchevski˜ [?]. See also Yanchevski˜ [?] for the relation between

± ±

K1 Spin and decomposability of involutions.

278 IV. ALGEBRAS OF DEGREE FOUR

CHAPTER V

Algebras of Degree Three

The main topic of this chapter is central simple algebras of degree 3 with

involutions of the second kind and their ´tale (commutative) subalgebras. To every

e

involution of the second kind on a central simple algebra of degree 3, we attach

a 3-fold P¬ster form, and we show that this quadratic form classi¬es involutions

up to conjugation. Involutions whose associated P¬ster form is hyperbolic form

a distinguished conjugacy class; we show that such an involution is present on

every central simple algebra of degree 3 with involution of the second kind, and we

characterize distinguished involutions in terms of ´tale subalgebras of symmetric