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    Irreducible Field Functions

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    Let F be a field and le p(x) E F[x]. If p(x) is not irreducible, then F[x]/{p(x)} is:

    a) always a field
    b) sometimes a field
    c) never a field.

    Give reasons for your assertion.

    Please see attached.

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    Solution Preview

    The answer is (c), never a field.

    Since p(x) is not irreducible, then it is reducible. We can find q(x) and r(x) with ...

    Solution Summary

    The irreducible field functions are examined. The reasons for the assertions are provided.