Prove that $\beta$ is a root of a $n$ degree polynomial of K [x] with leading coefficient $1$.

Question

Prove that $\beta$ is a root of a $n$ degree polynomial of K [x] with leading coefficient $1$. Here $K$ is UFD, $Q$ is its field of fractions, $Q(\beta)$ is algebraic extension of $n>1$ degree of the field of fractions, obtained by adjoining integral over $K$ element $\beta$.

Answer 1

It should be a consequence of Gauss' Lemma.

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That is, consider the monic polynomial $f$ of minimal degree in $K[x]$ which has $\beta$ as a root. This is clearly irreducible in $K[x]$, and thus, by Gauss' Lemma, also in $Q[x]$. So $f$ is the minimal polynomial of $\beta$ over $Q$, and thus $n = \lvert Q(\beta) : Q \rvert = \operatorname{degree(f)}$.