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Division algebras
Jan Vonk
1
Introduction.
In this short note, we will investigate the structure of a division algebra B over a ring A.
We will give an overview of some classification theorems, and give some arithmetically
important examples.
Definition. Let A be a commutative ring and B and A-module with a multiplication
· : B × B → B. Then B is a division algebra over A if B has a multiplicative identity
and every b ∈ B has a multiplicative inverse.
Rather than trying to work in full generality, we will consider specific rings A and
impose conditions on B. To simplify the discussion, we will always assume B is
finitely generated as an A-module. Even though many interesting structures arise
when this condition is omitted, this would lead us too far. Note that the most interesting cases occur when A is a field, since this gives B the structure of a finite dimensional
vector space. Many authors choose to include this in their definition of a division algebra. It should also be noted that many authors choose not to require B to have an
identity.
A = Z : Division algebras in full generality. Very little can be said in this case. We
will prove Wedderburn’s theorem and construct some important division algebras that
serve as examples later on.
A = R : The most interesting case. We will prove that the dimension of such a division algebra must be a power of 2. We classify all possible associative division algebras.
A = C : We prove that the only complex division algebra is C itself. The proofs in
this section apply without change when A is any algebraically closed field.
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2
Division algebras over Z.
Since every abelian group is a Z-module in a natural way, this is no real restriction
and we find ourselves in the most general case. If we assume that B is associative, we
can prove the following celebrated result:
Theorem 1 (Wedderburn). If B is associative and of finite order, then it is a field.
Proof. The center Z of B is a finite field, say |Z| = q, and B is a Z-vector space of
size q n . Considering the conjugation action of B × on itself, and counting the elements
of B × in every conjugacy class separately, we obtain
qn − 1 = q − 1 +
X qn − 1
qd − 1
,
where the sum is over all orbits of length ≥ 2, whose centralisers have sizes q d − 1 for
some d < n since they are also vector spaces over Z.
Now call Φn (x) the n-th cyclotomic polynomial, then Φn (q) divides every term in the
sum, as well as q n − 1. Hence also Φn (q) | q − 1. This is impossible since
Y 2πiλ n
|Φn (q)| =
> q − 1,
q − e
(λ,n)=1
♠
unless n = 1. Hence B must be a field.
Remark. Associativity is crucial here, since we made use of the group structure on
B × in an essential way by considering the action by conjugation on itself and using the
orbit-stabilizer formula.
Finally, we give some examples of division algebras. Naturally, our most interesting
examples will be the ones where A is a field.
• Finite fields Fq with q a prime power (as in the above theorem) are division
algebras. This is a rather dull example since any field is a priori a division
algebra.
• The quaternion algebra H: This is a 4-dimensional, real, associative division
algebra. We can define it as the four dimensional vector space over R with basis
2
{1, i, j, k} and multiplication determined by
·
1
i
j
k
1 i
j
k
1 i
j
k
i −1 k −j .
j −k −1 i
k j −i −1
There is a Euclidean norm on this algebra given by kak = a · a ∈ R, where · is the
automorphism flipping the signs of i, j, k. Notice now that H has multiplicative
a
.
identity 1 and any a ∈ H has inverse kak
• The octonion algebra O: This is an 8-dimensional, real, non-associative division
algebra. We can define it as the eight dimensional vector space over R with basis
{1, e1 , e2 , e3 , e4 , e5 , e6 , e7 } and multiplication table
·
1
e1
e2
e3
e4
e5
e6
e7
1 e1
e2
e3
e4
e5
e6
e7
1 e1
e2
e3
e4
e5
e6
e7
e1 −1 e3 −e2 e5 −e4 −e7 e6
e2 −e3 −1 e1
e6
e7 −e4 −e5
e3 e2 −e1 −1 e7 −e6 e5 −e4 .
e4 −e5 −e6 −e7 −1 e1
e2
e3
e5 e4 −e7 e6 −e1 −1 −e3 e2
e6 e7
e4 −e5 −e2 e3 −1 −e1
e7 −e6 e5
e4 −e3 −e2 e1 −1
The non-associativity can already be noted from the multiplication table. In a
similar fashion to the quaternions, we can define a multiplicative Euclidean norm
on the octonions as kak = a · a. This implies the existence of an inverse for any
element as before.
• As an example of a division algebra without a multiplicative identity (which
some authors allow) we have the additive group of complex numbers with the
multiplication C × C → C : (z, w) 7→ zw.
• Given a field k with characteristic 6= 2 and two nonzero elements a, b ∈ k, we can
define a quaternion algebra over k as the four dimensional vector space spanned
by {1, i, j, k} such that i2 = a, j 2 = b, ij = k, ji = −k. The algebras thus
obtained for k = Q are of great arithmetic importance.
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3
Division algebra’s over R.
The story of Hamilton’s revelation during a walk along the Royal Canal to the Royal
Irish Academy in 1843 belongs to mathematical folklore. Supposedly Hamilton had
been looking for a three dimensional division algebra over R for quite some time, before
he came up with a four dimenional one which is known as the algebra of Hamilton
quaterions these days. We will investigate division algebras over R is this section. If
we assume associativity, a full classification is known. In general, we will prove that
the dimension must necessarily be a power of 2. This curious fact gives an explanation
as to why Hamilton’s sharp mind failed to come up with an example.
3.1
Case I: B is associative.
In this case, we can give a full classification.
Theorem 2 (Frobenius). Every finite-dimensional associative division algebra D over
R is isomorphic to R, C or H.
The following proof is a combination of [1, Theorem 2B.5] and [3]. It uses techniques in
algebraic topology to dispose of the commutative case, and proceeds using a strategy
found in [3].
Proof. Assume D is of dimension n > 1. Suppose first that D is commutative. Then
2
f˜ : S n−1 → S n−1 : x 7→ kxx2 k , where the norm is just the usual Euclidean norm on
Rn , induces a well defined continuous map f : RP n−1 → S n−1 which we will prove
is injective. Assume that f (x) = f (y), then x2 − α2 y 2 = (x + αy)(x − αy) = 0 for
α2 = kx2 k / ky 2 k and hence x = ±αy, which implies α = ±1 by taking the norm of
both sides. Hence our map f is injective.
Since f is an injective map between compact Hausdorff spaces, it is a homeomorphism
onto its image. Since any embedding of compact n-manifolds is surjective, we must
have RP n−1 ∼
= S n−1 which can only happen if n = 2 by considering the homology
groups of both sides. This implies that D ∼
= C after some easy algebra.
For a general D, pick a nonzero element i ∈ D\R and consider Rhii := R ⊕ iR. This
is a two dimensional commutative division algebra over R and hence isomorphic to C.
Hence we can pick i such that i2 = −1, and hence justify our notation. Now consider
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D as a left C vector space, and call T the linear transformation on D of right multiplication by i. This operator has eigenvalues i and −i with corresponding eigenspaces
D+ and D− . Since x = 21 (x − ixi) + 21 (x + ixi) we even have D = D+ ⊕ D− .
Clearly D+ = C since if d ∈ D is in D+ then it commutes with C and the algebra generated by C and d would be of dimension strictly larger than 2 and yet commutative,
which is impossible as we showed earlier. Now take a nonzero element d ∈ D− , then
right multiplication by d clearly interchanges D+ and D− . Hence D− is a 1-dimensional
complex vector space, and D is a 4-dimensional real division algebra.
To finish up, note that d2 ∈ D− d = D+ and obviously d2 ∈ Rhdi, so d2 ∈ Rhdi ∩ D+ =
R. Furthermore d2 < 0 since otherwise it would have two square roots in R, on top of
square root d. This amounts to three different square roots in the field Rhdi, which is
impossible. Hence we can rescale d to an element j such that j 2 = −1. Defining k = ij
we can check all the identities to be satisfied, and clearly 1, i, j, k generate D.
♠
Remark. Let α be a root of f ∈ C[X] in some algebraic closure of C. Then C(α) is a
finite field extension of R, which by the above (since H is not commutative) must have
degree 2 over R. This proves the fundamental theorem of algebra.
Remark. Associativity is essential in the above proof. For example in stating that
D+ consists of all the elements commuting with i. As it turns out there are other
cases possible when we drop associativity. We have already seen the octonions, and
will investigate the situation more closely in the next section.
3.2
Case II: General B.
Keeping the constructions from the first section in mind, we have R, C, H and O (not
associative) as examples of division algebras over R. The dimensions of these algebras
as real vector spaces are 1, 2, 4 and 8. The following theorem confirms the obvious
conjecture the reader might make.
Theorem 3 (Hopf). The dimension of B over R must be a power of 2.
Proof. Setup: Call n the dimension of B. Then we can define the map
φ̃ : S n−1 × S n−1 → S n−1 : (a, b) 7→
5
ab
,
kabk
where we divide by the usual Euclidean norm on a real vector space to assure that
the map lands in S n−1 . One checks easily that this induces a well defined map φ :
RP n−1 × RP n−1 → RP n−1 , which in turn defines a ring morphism
φ∗ : H ∗ (RP n−1 ; F2 ) → H ∗ (RP n−1 × RP n−1 ; F2 ).
We will investigate this ring morphism in degree 1. Note that H 1 (RP n−1 ; F2 ) = F2 ,
and call γ the generator. Call α and β the pullbacks of γ under the natural projections
of RP n−1 × RP n−1 onto its two factors.
Step 1: We claim that φ∗ (γ) = α + β. When n ≤ 2, we have nothing to prove.
When n > 2, we know that π 1 (RP n−1 ) ∼
= F2 , generated by the reduction of any path
n−1
λ : [0, 1] → S
connecting two antipodal points x and −x. For fixed y we have that
[0, 1] → S n−1 : s 7→ φ̃(λ(s), y)
is a nontrivial loop in S n−1 , so h takes the nontrivial loop in the first factor of RP n−1 ×
RP n−1 to a nontrivial loop in RP n−1 . Similarly for the second factor. This shows what
we wanted to prove.
Step 2: Since H n (RP n−1 ; F2 ) = 0, we have γ n = (α + β)n = 0, in the ring H ∗ (RP n−1 ×
RP n−1 ; F2 ). By the Künneth formula from the appendix, this ring is isomorphic to
F2 [α, β]/(αn , β n ). Hence every
which is
term in the binomial expansion must vanish,
n
a
equivalent to saying
that
≡
0
(mod
2)
for
all
0
<
k
<
n.
Write
n
=
2
·
b,
with b
k
n
odd. Then 2a (b−1) is clearly odd, which implies that b = 1 and n is a power of 2.
♠
Remark. In fact, a much stronger result by Bott and Milnor proves that the dimension
has to be 1, 2, 4 or 8. We will omit the proof here for two reasons. The first being that
it would multiply the length of this note by a fairly large integer, and the second that
we do not have enough background to fully grasp the arguments ourselves. The reader
with more expertise is suggested to read [2].
Remark. We have no real hopes of adapting the above proof to get a bound on the
size of n, since the divisibility argument fails to see the size of n.
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4
Division algebra’s over C.
This is easy. The following theorem will imply that C is the only such example.
Theorem 4. Every division algebra over an algebraically closed field k must be isomorphic to k itself.
Proof. Take such a division algebra B, which must necessarily contain k. Let b ∈ B be
an element which is not in k, then the set {bn : n ∈ N} can not be linearly independent
since the dimension of B over k is finite. Hence there is a polynomial f (x) ∈ k[x]
such that f (b) = 0. Since k is algebraically closed, this implies that b ∈ k, hence the
result.
♠
Remark. Notice that the above proof does not need associativity of B. This is an
interesting feature of the algebraically closed case, which makes the notion of division
algebra rather superficial.
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Appendix: Results from topology.
4.1
The cup product.
The cohomology is derived from the dual of the chain complex for homology. Therefore, we do not expect more information encoded in the cohomology groups. One of
the reasons the definition of cohomology groups is still interesting, is the existence of
a ’multiplication’ on cohomology, the so called cup product. This raises many new
questions about the structure of the thus obtained ring H ∗ (X; R). In this section we
will discuss some of these structural results.
♣ First of all, we would like to know if this ring is commutative. As it turns out (see
[1, Theorem 3.14]) we have the following result.
Theorem 5. Let R be commutative. For all α ∈ H k (X; R) and β ∈ H l (X; R) we have
α ^ β = (−1)kl β ^ α.
As an example, we obtain commutativity when we assume R = F2 .
♣ Another fairly natural question to ask is how the ring structure of H ∗ (X × Y ; R)
relates to those of H ∗ (X; R) and H ∗ (Y ; R). An answer is given by the following
Künneth formula.
Theorem 6. If X and Y are CW-complexes and H k (Y ; R) is a free R-module of finite
rank for all k, then
H ∗ (X × Y ; R) ∼
= H ∗ (X; R) ⊗R H ∗ (Y ; R).
Remark. Note that many other Künneth formulae are available, and this is by no
means the strongest one. It will suffice for our purposes though. A proof of this
Künneth formula may be found in [1, Theorem 3.16], and will be left to the interested
reader.
♣ We now develop a tool that establishes a certain symmetry in (co)homology. While
this is not a result about the multiplicative structure of the cohomology ring, it will be
used to determine it in many special cases. This powerful tool is known as Poincaré
duality, and holds on a fairly broad class of topological spaces X.
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Theorem 7 (Poincaré duality). Let M be a closed R-orientable n-manifold with fundamental class [M ] ∈ Hn (M ; R), then the map
D : H k (M ; R) → Hn−k (M ; R) : α 7→ [M ] ∩ α
is an isomorphism, for all k.
For a proof, as well as definitions in the statement, the reader is refered to [1, Section
3.3]. We will be interested in the case R = F2 , rendering the condition of orientability
satisfied for any M .
4.2
The structure of H ∗ (RP n ; F2 ).
The cup product as described above gives a natural ring structure to H ∗ (X; R) for any
space X. In this section, we will use the previous results to determine the structure of
one particular cohomology ring that is relevant for the proofs in the main text.
Theorem 8. We have that
H ∗ (RP n ; F2 ) ∼
= F2 [x]/(xn+1 ).
Proof. As an easy argument using cellular homology and the universal coefficient theorem show (see, for example [1, Example 2.42]) we have that all the cohomology groups
H i are isomorphic to F2 for i ≤ n and 0 otherwise. The only question remaining is
how they patch together under the cup product.
We give an argument by induction on n, the case n = 1 being clear. For general n, we
have the inclusion RP n−1 → RP n . This induces isomorphisms of the H i for i < n, and
hence H i (RP n ; F2 ) = hαi i for α the generator of H 1 (RP n ; F2 ) ∼
= F2 .
The bilinear pairing defined by
H n−k (RP n ; F2 ) × H k (RP n ; F2 ) → F2 : (φ, ψ) 7→ (φ ^ ψ)[RP n ],
is perfect. Indeed, the induced map H n−k (RP n ; F2 ) → HomF2 (H k (RP n ; F2 ), F2 ) is obtained by composing the isomorphism from the universal coefficient theorem with the
dual of the Poincaré map. Note that Poincaré duality holds on the closed F2 -orientable
n-manifold RP n .
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Let φ : H n−1 (RP n ; F2 ) → F2 be the unique nontrivial morphism. Then this is realised
as taking the cup product with an element of H 1 (RP n ; F2 ) and evaluating at [RP n ].
This shows that mα ∪ αn−1 = mαn generates H n (RP n ; F2 ) for m = 0 or 1. Clearly
m = 1 so the generator of H n (RP n ; F2 ) is αn , which completes the proof.
♠
The following corollary is an essential ingredient for the proof of Hopf’s theorem in the
main text.
Corollary 1. We have that
H ∗ (RP n × RP n ; F2 ) ∼
= F2 [α, β]/(αn+1 , β n+1 ).
Proof. This follows straightforwardly using the above theorem and the Künneth formula 6.
♠
References
[1] A. Hatcher, Algebraic Topology, Cambridge University Press (2002).
[2] R. Bott and J. Milnor, On the Parallelizability of the Spheres, Bull. Amer. Math.
Soc. 64, 87-89, 1958.
[3] R.S. Palais, The Classification of Real Division Algebras , Vol. 75, No. 4 (1968),
pp. 366-368.
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