Division algebra

Summary

A division algebra is a nonzero algebra over a field in which one can divide by every nonzero element on both sides. The algebra need not be associative; an associative division algebra is a division ring that contains the field in its center. By the Frobenius theorem, the finite-dimensional associative ones over $\R$ are $\R$, $\C$ and the quaternions. The octonions are a nonassociative one. By the Bott–Milnor–Kervaire theorem, every finite-dimensional real division algebra has dimension $1$, $2$, $4$ or $8$.