Suppose that $P(x)\in\mathbb{Q}[x]$ is irreducible over $\mathbb{Q}$, and let $K$ be the $n$-th cyclotomic field. Is there a simple criterion to tell if $P$ remains irreducible over $K$? (Preferably a necessary and sufficient condition, unlike Eisenstein's.)
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A polynomial $P(x) \in \mathbb{Q}[x]$ is irreducible iff it is a prime element in the ring $\mathbb{Q}[x]$ iff $\mathbb{Q}[x]/(P(x))$ is a field, say $L$. To say that the polynomial remains irreducible in an extension $K/\mathbb{Q}$ is to say that $\mathbb{Q}[x]/(P(x)) \otimes_{\mathbb{Q}} K$ is a field, i.e., that the fields $K$ and $L$ are linearly disjoint over $\mathbb{Q}$. In your case, your extension $K/\mathbb{Q}$ is Galois, so you're in luck: if $K,L/\mathbb{Q}$ are two finite degree field extensions at least one of which is Galois, then linear disjointness is equivalent to $K \cap L = \mathbb{Q}$. So there's your necessary and sufficient condition: you want $L \cap \mathbb{Q}(\zeta_N) = \mathbb{Q}$. This is a condition that a mathematical software package will be able to check for given $P$ and $N$. I'm not a computational number theorist, but I would have to imagine that a computer will have a much easier time checking this than, say, factoring $P$ over $\mathbb{Q}(\zeta_N)$. |
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