Moreover, if char F = 2 we have x2 + x2 + x2 = 0, hence

1 2 3

x3 x3 + x 3 x3 + x 3 x3 p3 + q 2

x1 x2 x1 x3 x2 x3 12 13 23

+2 +2 = = .

x2 + x 2 x1 + x 2 x2 + x 2 x2 x2 x2 q2

1 2 3 3 123

Thus,

F [t]/(t2 + 4p3 + 27q 2 ) if char F = 2

∆(L)

and

F [t]/(t2 + t + 1 + p3 q ’2 )

∆(L) if char F = 2.

F [t]/ t2 + 3(a2 ’ 4) if f = X 3 ’ 3X + a and

In particular, we have ∆(L)

char F = 2, 3, and ∆(L) F [t]/(t2 + t + 1) if f = X 3 ’ b and char F = 3.

F [t]/(t2 + t + 1), then the form Q on ∆(L) de¬ned in

Conversely, if ∆(L)

(??) is isotropic: for j = t + (t2 + t + 1) ∈ ∆(L) we have T∆(L)/F (j) = ’1 and

N∆(L)/F (j) = 1, hence Q(j) = 0. Lemma (??) then shows that there is a nonzero

element x ∈ L0 such that S 0 (x) = 0. An inspection of the cases where L is not a

F [X]/(X 3 ’ b) where

¬eld shows that x is primitive in all cases. Therefore, L

b = x3 = NL/F (x).

Suppose ¬nally that char F = 3, and let δ ∈ F — be such that ∆(L) F [t]/(t2 ’

δ). The element d = t + (t2 ’ δ) ∈ ∆(L) then satis¬es

N∆(L)/F (d) ’ T∆(L)/F (d)2 = ’δ,

hence (??) shows that there exists x ∈ L0 such that S 0 (x) = ’δ. This element may

be chosen primitive, and its minimal polynomial then has the form

X 3 ’ δX + a for some a ∈ F .

A careful inspection of the argument in the proof above shows that if char F = 3

and L F — F — F one may ¬nd a ∈ F such that L F [X]/(X 3 ’ 3X + a) as

soon as card F ≥ 8. The same conclusion holds if L F — K for some quadratic

¬eld extension K when card F ≥ 4.

The group of square classes. Let L be a cubic ´tale F -algebra. The inclu-

e

sion F ’ L and the norm map NL/F : L ’ F induce maps on the square class

groups:

i : F — /F —2 ’ L— /L—2 , N : L— /L—2 ’ F — /F —2 .

Since NL/F (x) = x3 for all x ∈ F , the composition N —¦ i is the identity on F — /F —2 :

N —¦ i = IdF — /F —2 .

In order to relate L— /L—2 and F — /F —2 by an exact sequence, we de¬ne a map

#

: L ’ L as follows: for ∈ L we set

# 2

(18.33) = ’ TL/F ( ) + SL/F ( ) ∈ L,

so that

#

’ NL/F ( ) = 0.

In particular, for ∈ L— we have # = NL/F ( ) ’1 , hence # de¬nes an endomor-

phism of L— . We also put # for the induced endomorphism of L— /L—2 .

302 V. ALGEBRAS OF DEGREE THREE

(18.34) Proposition. The following sequence is exact:

#

i N

1 ’ F — /F —2 ’ L— /L—2 ’ L— /L—2 ’ F — /F —2 ’ 1.

’ ’ ’

Moreover, for x ∈ F — /F —2 and y ∈ L— /L—2 ,

(y # )# = i —¦ N (y) · y.

x = N —¦ i(x) and

Proof : It was observed above that N —¦ i is the identity on F — /F —2 . Therefore, i

is injective and N is surjective. For ∈ L— we have # = NL/F ( ) ’1 , hence

y # = i —¦ N (y) · y for y ∈ L— /L—2 .

(18.35)

Taking the image of each side under N , we obtain N (y # ) = 1, since N —¦ i —¦ N (y) =

N (y). Therefore, substituting y # for y in (??) we get (y # )# = y # = i —¦ N (y) · y.

In particular, if y # = 1 we have y = i —¦ N (y), and if N (y) = 1 we have y = y # .

Therefore, the kernel of # is in the image of i, and the kernel of N is in the image

of # . To complete the proof, observe that putting y = i(x) in (??) yields i(x)# = 1

for x ∈ F — /F —2 , since N —¦ i(x) = x.

§19. Central Simple Algebras of Degree Three

In this section, we turn to central simple algebras of degree 3. We ¬rst prove

Wedderburn™s theorem which shows that these algebras are cyclic, and we next

discuss their involutions of unitary type. It turns out that involutions of unitary

type on a given central simple algebra of degree 3 are classi¬ed up to conjugation

by a 3-fold P¬ster form. In the ¬nal subsection, we relate this invariant to cubic

´tale subalgebras and prove a theorem of Albert on the existence of certain cubic

e

´tale subalgebras.

e

19.A. Cyclic algebras. To simplify the notation, we set C3 = Z/3Z and

ρ = 1 + 3Z ∈ C3 .

Given a Galois C3 -algebra L over F and an element a ∈ F — , the cyclic algebra

(L, a) is de¬ned as follows:

(L, a) = L • Lz • Lz 2

where z is subject to the relations:

z3 = a

z = ρ( )z,

for all ∈ L.

(19.1) Example. Let L = F — F — F with the C3 -structure de¬ned by

ρ(x1 , x2 , x3 ) = (x3 , x1 , x2 ) for (x1 , x2 , x3 ) ∈ L.

M3 (F ) for all a ∈ F — . An explicit isomorphism is given by

We have (L, a)

« «

x1 00a

(x1 , x2 , x3 ) ’ and z ’ 1 0 0 .

x2

x3 010

From this example, it readily follows that (L, a) is a central simple F -algebra

for all Galois C3 -algebras L and all a ∈ F — , since (L, a) —F Fsep (L —F Fsep , a)

and L —F Fsep Fsep — Fsep — Fsep . Of course, this is also easy to prove without

extending scalars: see for instance Draxl [?, p. 49].

The main result of this subsection is the following:

§19. CENTRAL SIMPLE ALGEBRAS OF DEGREE THREE 303

(19.2) Theorem (Wedderburn). Every central simple F -algebra of degree 3 is

cyclic.

The proof below is due to Haile [?]; it is free from restrictions on the charac-

teristic of F . However, the proof can be somewhat simpli¬ed if char F = 3: see

Draxl [?, p. 63] or Jacobson [?, p. 80].

(19.3) Lemma. Let A be a central simple F -algebra of degree 3 and let x ∈ A — .

If Trd(x) = Trd(x’1 ) = 0, then x3 = Nrd(x).

Proof : If the reduced characteristic polynomial of x is

X 3 ’ Trd(x)X 2 + Srd(x)X ’ Nrd(x),

then the reduced characteristic polynomial of x’1 is

Srd(x) 2 Trd(x) 1

X3 ’ X+ X’ .

Nrd(x) Nrd(x) Nrd(x)

Therefore, Trd(x’1 ) = Srd(x) Nrd(x’1 ), and it follows that the reduced character-

istic polynomial of x takes the form X 3 ’ Nrd(x) if Trd(x) = Trd(x’1 ) = 0.

Proof of Theorem (??): In view of Example (??), it su¬ces to prove that central

division F -algebras of degree 3 are cyclic. Let D be such a division algebra. We

claim that it su¬ces to ¬nd elements y, z ∈ D such that z ∈ F , y ∈ F (z) and

Trd(x) = Trd(x’1 ) = 0 for x = z, yz, yz 2 . Indeed, the lemma then shows

z 3 = Nrd(z), (yz)3 = Nrd(yz), (yz 2 )3 = Nrd(yz 2 );

since Nrd(yz 2 ) = Nrd(yz) Nrd(z) it follows that

(yz 2 )3 = (yz)3 z 3

hence, after cancellation,

zyz 2y = yzyz 2.

By dividing each side by z 3 = Nrd(z), we obtain

(zyz ’1)y = y(zyz ’1),

which shows that zyz ’1 ∈ F (y). We have zyz ’1 = y since y ∈ F (z), hence we may

de¬ne a Galois C3 -algebra structure on F (y) by letting ρ(y) = zyz ’1; then,

D F (y), Nrd(z) .

We now proceed to construct elements y, z satisfying the required conditions.

Let L ‚ D be an arbitrary maximal sub¬eld. Considering D as a bilinear

space for the nonsingular bilinear form induced by the reduced trace, pick a nonzero

element u1 ∈ L⊥ . Since dim (u’1 F )⊥ ©L ≥ 2, we may ¬nd u2 ∈ (u’1 F )⊥ ©L such

1 1

that u2 ∈ u1 F . Set z = u1 u2 . We have Trd(z) = 0 because u1 ∈ L and u’1 ∈ L,

’1 ⊥

2

and Trd(z ’1 ) = 0 because u2 ∈ (u’1 F )⊥ . Moreover, z ∈ F since u2 ∈ u1 F .

1

Next, pick a nonzero element v1 ∈ F (z)⊥ F (z). Since dim(zF + z ’1 F )⊥ = 7,

we have

v1 (zF + z ’1 F )⊥ © F (z) = {0};

’1

we may thus ¬nd a nonzero element v2 in this intersection. Set y = v2 v1 . Since

v1 ∈ F (z), we have y ∈ F (z). On the other hand, since v1 ∈ F (z)⊥ and v2 ∈ F (z),

we have

’1 ’1

Trd(yz 2 ) = Trd(v1 z 2 v2 ) = 0.

Trd(yz) = Trd(v1 zv2 ) = 0 and

304 V. ALGEBRAS OF DEGREE THREE

’1

Since v1 v2 ∈ (zF + z ’1 F )⊥ and z 3 = Nrd(z), we also have

’1

Trd(z ’1 y ’1 ) = Trd(z ’1 v1 v2 ) = 0,

’1

Trd(z ’2 y ’1 ) = Nrd(z) Trd(zv1 v2 ) = 0.

The elements y and z thus meet our requirements.

19.B. Classi¬cation of involutions of the second kind. Let K be a quad-

ratic ´tale extension of F , and let B be a central simple28 K-algebra of degree 3

e

such that NK/F (B) is split. By Theorem (??), this condition is necessary and su¬-

cient for the existence of involutions of the second kind on B which ¬x the elements

of F .

We aim to classify those involutions up to conjugation, by means of the as-

sociated trace form on Sym(B, „ ) (see §??). We therefore assume char F = 2

throughout this subsection.

(19.4) De¬nition. Let „ be an involution of the second kind on B which is the

identity on F . Recall from §?? the quadratic form Q„ on Sym(B, „ ) de¬ned by

Q„ (x) = TrdB (x2 ) for x ∈ Sym(B, „ ).

Let ± ∈ F be such that K F [X]/(X 2 ’ ±). Proposition (??) shows that Q„ has

a diagonalization of the form:

Q„ = 1, 1, 1 ⊥ 2 · ± · q„

where q„ is a 3-dimensional quadratic form of determinant 1. The form

π(„ ) = ± · q„ ⊥ ±

is uniquely determined by „ up to isometry and is a 3-fold P¬ster form since det q„ =

∼

1. We call it the P¬ster form of „ . Every F -isomorphism (B, „ ) ’ (B , „ ) of

’

∼

algebras with involution induces an isometry of trace forms Sym(B, „ ), Q„ ’ ’

∼

Sym(B , „ ), Q„ , hence also an isometry π(„ ) ’ π(„ ). The P¬ster form π(„ ) is

’

therefore an invariant of the conjugacy class of „ . Our main result (Theorem (??))

is that it determines this conjugacy class uniquely.

(19.5) Example. Let V be a 3-dimensional vector space over K with a nonsingular

hermitian form h. Let h = δ1 , δ2 , δ3 K be a diagonalization of this form (so that

δ1 , δ2 , δ3 ∈ F — ) and let „ = „h be the adjoint involution on B = EndK (V ) with

respect to h.

Propositions (??) and (??) show that

Q„ 1, 1, 1 ⊥ 2 · ± · δ1 δ2 , δ1 δ3 , δ2 δ3 .

Therefore,

π(„ ) = ± · ’δ1 δ2 , ’δ1 δ3 .

(19.6) Theorem. Let B be a central simple K-algebra of degree 3 and let „ , „

be involutions of the second kind on B ¬xing the elements of F . The following

conditions are equivalent:

(1) „ and „ are conjugate, i.e., there exists u ∈ B — such that

„ = Int(u) —¦ „ —¦ Int(u)’1 ;

28 We A — Aop for some central simple F -algebra A of

allow K F — F , in which case B

degree 3.

§19. CENTRAL SIMPLE ALGEBRAS OF DEGREE THREE 305

(2) Q„ Q„ ;

(3) π(„ ) π(„ ).

Proof : (??) ’ (??) Conjugation by u de¬nes an isometry from Sym(B, „ ), Q„

to Sym(B, „ ), Q„ .

(??) ” (??) This follows by Witt cancellation, in view of the relation between

Q„ and π(„ ).

(??) ’ (??) If K F — F , then all the involutions on B are conjugate to

the exchange involution. We may therefore assume K is a ¬eld. If B is split, let

B = EndK (V ) for some 3-dimensional vector space V . The involutions „ and „

are adjoint to nonsingular hermitian forms h, h on V . Let

h = δ 1 , δ2 , δ3 and h = δ1 , δ2 , δ3

K K

(with δ1 , . . . , δ3 ∈ F — ) be diagonalizations of these forms. As we observed in (??),

we have

Q„ 1, 1, 1 ⊥ 2 · ± · δ1 δ2 , δ1 δ3 , δ2 δ3

and

Q„ 1, 1, 1 ⊥ 2 · ± · δ1 δ2 , δ1 δ3 , δ2 δ3 .

Therefore, condition (??) implies

± · δ 1 δ2 , δ1 δ3 , δ2 δ3 ± · δ 1 δ2 , δ1 δ3 , δ2 δ3 .

It follows from a theorem of Jacobson (see Scharlau [?, Theorem 10.1.1]) that the

hermitian forms δ1 δ2 , δ1 δ3 , δ2 δ3 K and δ1 δ2 , δ1 δ3 , δ2 δ3 K are isometric; then

δ 1 δ2 δ3 · h δ 1 δ2 , δ1 δ3 , δ2 δ3 δ 1 δ2 , δ1 δ3 , δ2 δ3 δ 1 δ2 δ3 · h ,

K K

hence the hermitian forms h, h are similar. The involutions „ , „ are therefore

conjugate. This completes the proof in the case where B is split.

The general case is reduced to the split case by an odd-degree scalar extension.

Suppose B is a division algebra and let „ = Int(v) —¦ „ for some v ∈ B — , which may

be assumed symmetric under „ . By substituting Nrd(v)v for v, we may assume

Nrd(v) = µ2 for some µ ∈ F — . Let L/F be a cubic ¬eld extension contained in B.

(For example, we may take for L the sub¬eld of B generated by any noncentral

„ -symmetric element.) The algebra BL = B —F L is split, hence the argument

given above shows that „L and „L are conjugate:

—

„L = Int(u) —¦ „L —¦ Int(u)’1 = Int u„L (u) —¦ „L for some u ∈ BL ,

hence

v = »u„L (u) for some » ∈ KL = K —F L.

Since „ (v) = v, we have in fact » ∈ L— . Let ι be the nontrivial automorphism

of KL /L. Since Nrd(u) = µ2 , by taking the reduced norm on each side of the

preceding equality we obtain:

µ2 = »3 Nrd(u) · ι Nrd(u) ,

—

hence » = µ»’1 Nrd(u) · ι µ»’1 Nrd(u) and, letting w = µ»’1 Nrd(u)u ∈ BL ,

v = w„L (w).

By (??), there exists w ∈ B — such that v = w „ (w ). Therefore,

„ = Int(w ) —¦ „ —¦ Int(w )’1 .

306 V. ALGEBRAS OF DEGREE THREE

(19.7) Remark. If (B, „ ) and (B , „ ) are central simple K-algebras of degree 3

with involutions of the second kind leaving F elementwise invariant, the condition

π(„ ) π(„ ) does not imply (B, „ ) (B , „ ). For example, if K F — F all the

forms π(„ ) are hyperbolic since they contain the factor ± , but (B, „ ) and (B , „ )

are not isomorphic if B B .

(19.8) De¬nition. An involution of the second kind „ on a central simple K-

algebra B of degree 3 is called distinguished if π(„ ) is hyperbolic. Theorem (??)

shows that the distinguished involutions form a conjugacy class. If K F — F , all

involutions are distinguished (see the preceding remark).

(19.9) Example. Consider again the split case B = EndK (V ), as in (??). If

the hermitian form h is isotropic, then we may ¬nd a diagonalization of the form

h = 1, ’1, » K for some » ∈ F — , hence the computations in (??) show that the

adjoint involution „h is distinguished. Conversely, if h is a nonsingular hermitian

form whose adjoint involution „h is distinguished, then „h and „h are conjugate,

hence h is similar to h. Therefore, in the split case the distinguished involutions

are those which are adjoint to isotropic hermitian forms.

We next characterize distinguished involutions by a condition on the Witt index

of the restriction of Q„ to elements of trace zero. This characterization will be used

to prove the existence of distinguished involutions on arbitrary central simple K-

algebras B of degree 3 such that NK/F (B) is split, at least when char F = 3.

Let

Sym(B, „ )0 = { x ∈ Sym(B, „ ) | TrdB (x) = 0 }

and let Q0 be the restriction of the bilinear form Q„ to Sym(B, „ )0 . We avoid

„

the case where char F = 3, since then Q0 is singular (with radical F ). Assuming

„

char F = 2, 3, let w(Q0 ) be the Witt index of Q0 .

„ „

(19.10) Proposition. Suppose char F = 2, 3. The following conditions are equiv-

alent:

(1) „ is distinguished;

(2) w(Q0 ) ≥ 2;

„

(3) w(Q0 ) ≥ 3.

„

√

Proof : Let K = F ( ±). The subspace Sym(B, „ )0 is the orthogonal complement

of F in Sym(B, „ ) for the form Q„ ; since Q„ (1) = 3, it follows that Q„ = 3 ⊥ Q0 .

„

Since 1, 1, 1 3, 2, 6 , it follows from (??) that

Q0 2· 1, 3 ⊥ ± · q„ .

„

Since π(„ ) = ± · q„ ⊥ ± , we have

Q0 = 2 · 3, ± + π(„ ) in W F .

„

By comparing dimensions on each side, we obtain

w Q0 = w 3, ± ⊥ π(„ ) ’ 1.

„

Condition (??) implies that w 3, ± ⊥π(„ ) ≥ 4, hence the equality above yields (??).

On the other hand, condition (??) shows that w 3, ± ⊥ π(„ ) ≥ 3, from which

it follows that π(„ ) is isotropic, hence hyperbolic. Therefore, (??) ’ (??). Since

(??) ’ (??) is clear, the proof is complete.

§19. CENTRAL SIMPLE ALGEBRAS OF DEGREE THREE 307

The relation between TrdB and SrdB (see (??)) shows that for x ∈ Sym(B, „ )0 ,

the condition that TrdB (x2 ) = 0 is equivalent to SrdB (x) = 0. Therefore, the totally

isotropic subspaces of Sym(B, „ )0 for the form Q0 can be described as the subspaces

„

3

consisting of elements x such that x = NrdB (x). We may therefore reformulate

the preceding proposition as follows:

(19.11) Corollary. Suppose char F = 2, 3. The following conditions are equiva-

lent:

(1) „ is distinguished;

there exists a subspace U ‚ Sym(B, „ )0 of dimension 2 such that u3 = NrdB (u)

(2)

all u ∈ U ;

for

there exists a subspace U ‚ Sym(B, „ )0 of dimension 3 such that u3 = NrdB (u)

(3)

all u ∈ U .

for

We now prove the existence of distinguished involutions:

(19.12) Proposition. Suppose29 char F = 2, 3. Every central simple K-algebra B

such that NK/F (B) is split carries a distinguished involution.

Proof : In view of (??), we may assume that B is a division algebra. Let „ be

an arbitrary involution of the second kind on B which is the identity on F . Let

L ‚ Sym(B, „ ) be a cubic ¬eld extension of F and let u be a nonzero element in

the orthogonal complement L⊥ of L for the quadratic form Q„ . We claim that the

involution „ = Int(u) —¦ „ is distinguished.

In order to prove this, consider the F -vector space

U = L © (u’1 F )⊥ · u’1 .

Since Q„ is nonsingular, we have dim U ≥ 2. Moreover, since L ‚ Sym(B, „ ), we

have U ‚ Sym(B, „ ). For x ∈ L©(u’1 F )⊥ , x = 0, we have Trd(xu’1 ) = 0 because

x ∈ (u’1 F )⊥ , and Trd(ux’1 ) = 0 because u ∈ L⊥ and x’1 ∈ L. Therefore, for all

nonzero y ∈ U we have

Trd(y) = Trd(y ’1 ) = 0,

hence also y 3 = Nrd(y), by (??). Therefore, Corollary (??) shows that „ is distin-

guished.

(19.13) Remark. If char F = 3, the proof still shows that for every central simple

K-algebra B such that NK/F (B) is split, there exists a unitary involution „ on B

and a 2-dimensional subspace U ‚ Sym(B, „ ) such that u3 ∈ F for all u ∈ U .

´

19.C. Etale subalgebras. As in the preceding subsection, we consider a cen-

tral simple algebra B of degree 3 over a quadratic ´tale extension K of F such that

e

NK/F (B) is split. We continue to assume char F = 2, 3, and let ι be the nontrivial

automorphism of K/F . Our aim is to obtain information on the cubic ´tale F -

e

algebras L contained in B and on the involutions of the second kind which are the

identity on L.

29 See (??) for a di¬erent proof, which works also if char F = 3.

308 V. ALGEBRAS OF DEGREE THREE

Albert™s theorem. The ¬rst main result is a theorem of Albert [?] which

asserts the existence of cubic ´tale F -subalgebras L ‚ B with discriminant ∆(L)

e

isomorphic to K. For such an algebra, we have LK L — ∆(L) Σ(L), by (??),

hence LK can be endowed with a Galois S3 -algebra structure. This structure may

be used to give an explicit description of (B, „ ) as a cyclic algebra with involution.

(19.14) Theorem (Albert). Suppose30 char F = 2, 3 and let K be a quadratic

´tale extension of F . Every central simple K-algebra B such that N K/F (B) is split

e

contains a cubic ´tale F -algebra L with discriminant ∆(L) isomorphic to K.

e

Proof : (after Haile-Knus [?]). We ¬rst consider the easy special cases where B is

not a division algebra. If K F — F , then B A — Aop for some central simple F -

algebra A of degree 3. Wedderburn™s theorem (??) shows that A contains a Galois

C3 -algebra L0 over F . By (??), we have ∆(L0 ) F — F , hence we may set

0

L = {( , ) | ∈ L0 }.

If K is a ¬eld and B is split, then we may ¬nd in B a subalgebra L isomorphic to

F — K. Identifying B with M3 (K), we may then choose