Non-monic minimal polynomial implies element is not integral

Question

I know that the minimal polynomial of $\frac{1}{\sqrt{3}+1}$ over $\mathbb{Z}$ is given by $2X^2+2X-1$. It is obviously not monic. Can I immediately conclude that the element is not integral over $\mathbb{Z}$ ? Integrality requires a monic polynomial annihilating the element. Thanks for your help !

Answer 1

Yes, you can. If $f$ is monic having the given number as a root, then $2x^2+2x−1\mid f$ in $\mathbb Q[x]$. Then there is a non-zero integer $m$ such that $mf=(2x^2+2x−1)g$ with $g∈\mathbb Z[x]$. Now take the content on both sides and conclude that $m=c(g)$. Dividing by $m$ we get $f=(2x^2+2x−1)g_1$ in $\mathbb Z[x]$ which is impossible.