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    Proving that a Function is Irreducible

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    Let p be a prime congruent to -1 mod 4. Show that X^2 + 1 is irreducible in Z_p[X], and hence K = Z_p[X] / (X^2 + 1) is the field of order p^2. Note that K has a multiplication similar to that of the complex numbers.

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    Solution Preview

    Since p = -1 (mod 4), then -1 is not a quadratic residue of p. It means that the equation ...

    Solution Summary

    In this solution, we provide proof that the given function is irreducible.