How do I show there are irreducible polynomials of any degree in $\mathbb{Z}_p[x]$, with $p$ prime?
I tried counting the number of reducible polynomials of any degree but that turned out to be hard... Any help?
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You are on the right track. You can count the monic polynomials of degree $n$ in $\mathbb{Z}_p[X]$ and subtract the ones that have a factor of degree at most $n/2$. What remains is the number of irreducible polynomials there of degree $n$. – hardmath Aug 13 '15 at 00:27
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Wouldn't I be double counting? – user261688 Aug 13 '15 at 00:34
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4See Existence of irreducible polynomials over finite field, and Existence of an irreducible polynomial over $\mathbb{F}_p$, and Lack of understanding of the proof of the existence of an irreducible polynomial of any degree $n \ge 2$ in $\mathbb{Z}_p[x]$. – hardmath Aug 13 '15 at 00:38
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I cannot understand the third link and the rest uses things I haven't proven yet. Could you explain what you meant by counting? – user261688 Aug 13 '15 at 00:42
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Have a look at Find the number of irreducible polynomials in any given degree, which tackles the difficulty of exactly counting the irreducible monic polynomials in $\mathbb{Z}_p[X]$ of degrees 2 and 3. However an exact count is not required in order to prove existence; we only need a positive count! – hardmath Aug 13 '15 at 00:48
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Wait but I already tried this, you need to keep track of the amount of irreducibles of any degree and it doesn't work out nicely... – user261688 Aug 13 '15 at 01:01
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2Do you know the fact that $X^{p^n} - X$ is the product of all irreducible polynomials over $\mathbb{Z}_p$ whose degree divides $n$? This Answer by Marc van Leeuwen leverages that fact to show at least one irreducible polynomial of degree exactly $n$ exists. – hardmath Aug 13 '15 at 01:38