Intereting Posts

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Integral $\int{\sqrt{25 – x^2}dx}$
Applications of $Ext^n$ in algebraic geometry
a probability question using percentages
Prove that for all odd integer $n, n^{4}=1\pmod {16}$
Conformal mapping $z+\frac{1}{z}$, how to see the mapping to hyperbolas?
Surface under $\frac{1}{x}$ is $\infty$, while surface under $\frac{1}{x^2}$ is $1$?
Compact $n$-manifold has same integral cohomology as $S^n$?

We have the definition of integral closure that all the integral elements of *A* in *B*. Could we just compute the integral closure of certain *A* in *B*. I am considering such a problem that given a prime p, what is the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[x]/(x^n–p)$.

After some trials, I find the answer maybe $\mathbb{Z}[x]/(x^n–p)$. Since $\mathbb{Z}\subset\mathbb{Z}[x]/(x^n–p)$, the integral closure of $\mathbb{Z}$ should be the subset of the closure of $\mathbb{Z}[x]/(x^n–p)$. But $\mathbb{Z}[x]/(x^n–p)$ is generated by $1, s, s^2, \dots, s^{n–1}$ which are integral over $\mathbb{Z}$. $s$ is the image of x modulo (x^n–p). Therefore $\mathbb{Z}[x]/(x^n–p)$ is integral over $\mathbb{Z}$.

Now I want to prove that $\mathbb{Z}[x]/(x^n–p)$ is normal.

Assume that $u\in\mathbb{Q}[x]/(x^n–p)$ is integral over $\mathbb{Z}[x]/(x^n–p)$ with a monic poly $f$. Since $\exists d \in\mathbb{Z}$ and $e\in \mathbb{Z}[x]/(x^n–p)$ s.t du–e=0. Then f can be divided by dx-e. Assume that $f=(dx-e)g$, where $g$ is a poly with leading coefficient h in $\mathbb{Z}[x]/(x^n–p)$.

Then we get $hd=1$. Since $d$ is a integer, $h$ has to be a integer too. This implies that $d=1$ or $–1$. Both sides implies $u\in \mathbb{Z}[x]/(x^n–p)$. So we got the integral closure of $\mathbb{Z}$ contains $\mathbb{Z}[x]/(x^n–p)$ which is exactly the closure of itself. Q.E.D

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Is there any mistakes in the proof?

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