I am a little confused about terminology when it comes to extension fields. In my textbook, E is a field extension of F if F is a subfield of E. This is understandable. However, in proving that all polynomials have a zero in an extension field, ##F[x] / \langle p(x) \rangle##, where ##p(x)## is irreducible, is identified as an extension field of ##F##. But how does that match the definition of extension field given above? ##F## isn't a subfield of ##F[x] / \langle p(x) \rangle## at all, but rather isomorphic to a subfield by the isomorphism ##\mu (a) = a + \langle p(x) \rangle##, right?(adsbygoogle = window.adsbygoogle || []).push({});

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# I Extension fields

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