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Math Help - Irreducible polynomial mod p^2

  1. #1
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    Irreducible polynomial mod p^2

    Consider the polynomial x^{2} + p modulo p^{2}.

    I would like to show that this polynomial is irreducible. Unfortunately I don't see how I can apply any of the usual tricks (Eisenstein, simple enumeration in the case of fields of small characteristic) might work.
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  2. #2
    Senior Member roninpro's Avatar
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    Suppose x^2+p\pmod{p^2} is reducible. Then, x^2+p\equiv 0\pmod{p^2} has a solution. By reducing the modulus, we have x^2\equiv 0\pmod{p}, so x is divisible by p. Write x=py. Putting this back into the original equation gives (py)^2+p\equiv p^2y^2+p\equiv p \equiv 0\pmod{p^2}, which is certainly not true.

    Therefore, x^2+p\pmod{p^2} is irreducible.
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