Recently, a problem in my research has appeared and now I need to construct some algebraic numbers with special properties (related to its degree and some other fields extensions).


Now, in order to help me, I would like to prove the following result:


**Proposition.** *Let $\alpha$ be a real algebraic number and let $n>4$ be a positive integer, then the polynomial $X^n-p$ is irreducible over $K:=\mathbb{Q}(\alpha)$, for all large enough prime number $p$.*


I tried to use some splitting fields, discriminantes properties, (un)ramified primes to prove it, but I was not able to do it.


Any suggestion is very welcomed.


Thanks in advance.



Irreducibility of polynomials over some number fields
**Lemma.** *Let $K$ be any number field, and $p$ a prime unramified in $K$. Then $X^n-p$ is irreducible over $K$.*


*Proof.* It suffices to show that the field $L = K(\sqrt[n\ \ ]{p})$ has degree $n$ over $K$. Let $\mathfrak q \subseteq \mathcal O\_L$ be a prime above $p$, and let $\mathfrak p = \mathcal O\_K \cap \mathfrak q$ be its image in $\operatorname{Spec} \mathcal O\_K$. Since $\mathcal O\_L$ contains $\mathbf Z[\sqrt[n\ \ ]{p}]$, we have $e\_{\mathfrak q/p} \geq n$. But $K$ is unramified above $p$, so $e\_{\mathfrak p/p} = 1$. We conclude that $e\_{\mathfrak q/\mathfrak p} = n$, so $[L:K] \geq n$. The reverse inequality is clear. $\square$


