This is important in Galois theory, where the distinction is made between separable field extensions ( defined by polynomials with no multiple roots ) and inseparable ones.
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There is a classification theorem for separable algebras : separable algebras are the same as finite products of matrix algebras over finite-dimensional division algebras whose centers are finite-dimensional separable field extensions of the field " K ".
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Conversely, the primitive element theorem states that any finite separable field extension " L " / " K " can be generated by a single element " ? " " " L " and the preceding theory then gives a concrete description of the field " L " as the quotient of the polynomial ring " K " [ " X " ] by a principal ideal generated by an irreducible polynomial " p ".
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