Consider the polynomial modulo .

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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- Nov 19th 2010, 06:33 AMrobeulerIrreducible polynomial mod p^2
Consider the polynomial modulo .

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. - Nov 19th 2010, 07:54 AMroninpro
Suppose is reducible. Then, has a solution. By reducing the modulus, we have , so is divisible by . Write . Putting this back into the original equation gives , which is certainly not true.

Therefore, is irreducible.