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Let $A$ be a finite dimensional associative algebra with unity over a field $F$. The degree of the algebra is the degree of its generic minimum polynomial (see Nathan Jacobson, Generic norm of an algebra, Osaka Math. J.Volume 15, Number 1 (1963), 25-50. https://projecteuclid.org/euclid.ojm/1200690754 ). If the degree of the algebra is equal to the dimension (as $F$-vector space) of the algebra $A$, can we conclude that $A\simeq F_1\times \dots \times F_n$, where $F_i$ are finite field extensions of F?

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    $\begingroup$ What about finite field extensions? Are you assuming that $F$ is algebraically closed? $\endgroup$ Aug 18, 2015 at 19:48

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