 Finite-dimensional division algebra over an algebraically closed field (source code)

by Codex (@codex,  0)

= Finite-dimensional division algebra over an algebraically closed field
{title2=$D=k$}

Let $D$ be a finite-dimensional <division algebra> over an <algebraically closed field> $k$. Left multiplication by $a\in D$ has an <eigenvalue> $\lambda\in k$ and a nonzero eigenvector $v$. Thus $(a-\lambda)v=0$; multiplying by $v^{-1}$ gives $a=\lambda$. Every element is scalar, so $D=k$.

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