I am a undergradute math student and I was traying to prove the polynomial $p(x)= x^4-10x^2+1$ is irreducible in $\mathbb{Q}[x]$. I used reduction mod p and for every prime p I found a factorizarion in $\mathbb{Q}[x]$. I know $p(x)$ is irreducible, but i have a question about this method. If I try with many primes and anyone works, i have to try until I find one that works? or try with another method.Thanks for any comments.
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No, it never works for any prime here with the reduction criterion, but it is well known that the polynomial is irreducible over $\Bbb Q$. See this duplicate and this one. – Dietrich Burde Dec 13 '23 at 19:25
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Yes, but i have another question. If i have a polynomial p(x) and I try with reduction mod p, and if i try with p=2, and does not work, the i try with p=3, and does not work again, it could be posible that it works for p = 53, for example? Or, if with p=2,3 does not work, is recommendable try with other primes or just change the method? – Benjamín Garcés Dec 14 '23 at 01:10
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Yes, this can happen. We have many posts here as examples. See, say, this one. The polynomial is reducible modulo $p$ for $p=2,3,5,7$, but then irreducible for $p=11$. – Dietrich Burde Dec 14 '23 at 09:27