The Significance of Complex Numbers: Frobenius Theorem

By Xah Lee. Date: 2013-09-25

Learned a new thing today. This answered me indirectly a important question, namely, why are complex numbers special (among algebras). It also helped in understanding the relative importance of quaternions.

Frobenius theorem (real division algebras)

In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. (Division algebra is a algebra over a field where division is possible.) According to the theorem, every such algebra is isomorphic to one of the following:

ℝ (the real numbers)

ℂ (the complex numbers)

ℍ (the quaternions).

These algebras have dimensions 1, 2, and 4, respectively. Of these three algebras, the real and complex numbers are commutative, but the quaternions are not.

This theorem is closely related to Hurwitz's theorem, which states that the only normed division algebras over the real numbers are ℝ, ℂ, ℍ, and the (non-associative) algebra 𝕆 of octonions.

Hurwitz's theorem is a theorem of Adolf Hurwitz (1859 to 1919), published posthumously in 1923, on finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions or the octonions. The non-associative algebras occurring are called Hurwitz algebras or composition algebras. The problem has an equivalent formulation in terms of quadratic forms q(x), composability requiring the existence of a bilinear “composition” z(x, y) such that q(x) q(y) = q(z (x, y)). Subsequent proofs have used the Cayley–Dickson construction. Although neither commutative nor associative, composition algebras have the special property of being alternative algebras, i.e. left and right multiplication preserves squares, a weakened version of associativity. The theory has subsequently been generalized to arbitrary quadratic forms and arbitrary fields.[1]

Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898.