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When ab ≠ ba: division rings and the quaternions

Your first noncommutative world. We define division rings, build Hamilton's quaternions by hand, and meet the surprise that finite division rings are secretly fields.

Dropping commutativity

In Volume I a field was a commutative ring with 1 in which every nonzero element is invertible. Drop the word commutative and you get a division ring (also called a skew field): a ring with 1 ≠ 0 in which every nonzero element has a two-sided multiplicative inverse, but where ab and ba may differ. Fields are exactly the commutative division rings. The whole subject begins with the question: are there others?

The honest answer is yes, but they are scarce and precious. Over the real numbers there is essentially one noncommutative division ring you can build with finite dimension: the quaternions. That scarcity is a theorem (Frobenius's), not an accident, and it tells you how rigid this world is once you fix a center.

Building the quaternions

Hamilton's quaternions H form a 4-dimensional real vector space with basis 1, i, j, k. Multiplication is forced by three rules and bilinearity. Here is the full multiplication table — notice it is not symmetric across the diagonal, which is exactly the failure of commutativity.

Defining relations:   i^2 = j^2 = k^2 = -1,   ij = k,   ji = -k

Full table (row * column):
        1     i     j     k
  1     1     i     j     k
  i     i    -1     k    -j
  j     j    -k    -1     i
  k     k     j    -i    -1

So ij = k but ji = -k:  ij - ji = 2k  ≠ 0.  Order matters.

Inverse of q = a + bi + cj + dk:  use the conjugate q* = a - bi - cj - dk.
  q q* = a^2 + b^2 + c^2 + d^2 = N(q)  (a nonneg real),
  so for q ≠ 0,   q^(-1) = q* / N(q).
Every nonzero q is invertible  =>  H is a division ring.
The quaternion multiplication table and the conjugate trick that makes H a division ring.

Finite means commutative

Here is the first genuinely startling theorem of the subject. Wedderburn's little theorem: every finite division ring is a field. Noncommutative division rings exist, but never finitely many elements. The quaternions over the reals are infinite for a reason — over a finite field there simply is no room to escape commutativity.

The proof is a gem: combine the class equation of the multiplicative group D* with the algebra of cyclotomic polynomials. The center Z is a field with q elements; D has dimension n over Z, so |D*| = q^n − 1, and each centralizer contributes a factor q^d − 1 with d | n. The cyclotomic factor Φ_n(q) divides q^n − 1 and each proper q^d − 1, forcing |Φ_n(q)| ≤ q − 1, which is impossible for n > 1. Hence n = 1 and D = Z is commutative.