number fields - Given an integer n and a finite extension K of Q , find a polynomial of degree n that is irreducible over K

Here is a more general result.

Theorem: Let $(K,| |)$ be a non-Archimedean normed field with completion $hat{K}$. Let $mathcal{L}/hat{K}$ be a finite separable extension of degree $d$. Then there exists a degree $d$ separable field extension $L/K$ such that $Lhat{K} = mathcal{L}$.

In particular, as long as the completion of $K$ admits a separable field extension of a certain degree $d$, so does $K$ itself, necessarily of the form $K[t]/(P(t))$ by the primitive element theorem. Moreover, as long as $K$ has characteristic zero and carries a nontrivial discrete valuation, it admits finite separable extensions of all finite degrees.

For a proof of this theorem using Krasner's Lemma, see Section 3.5 of

http://math.uga.edu/~pete/8410Chapter3.pdf

When the norm corresponds to discrete valuation $v$ (e.g. $| | = | |_{mathfrak{p}}$
the $mathfrak{p}$-adic norm for a prime ideal $mathfrak{p}$ of a number field $K$) one can get away with less: by weak approximation, there exists $alpha in K$ with $v(alpha) = 1$. For any positive integer $n$ prime to the characteristic of $K$, by Eisenstein's Criterion the polynomial $t^n - alpha in K[t]$ is (separable and) irreducible even over the completion $hat{K}$, so is certainly irreducible over $K$.

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