Splitting field of $x^{p^{n}}-1\in\mathbb{Z}_{p}\left[x\right]$

Question

I'm trying to find the splitting field of $x^{p^{n}}-1\in\mathbb{Z}_{p}\left[x\right]$ where $p$ is a prime and integer $n\geq1$.

If $\alpha_{1},\dots,\alpha_{p^{n}}$ are the roots then $\mathbb{Z}_{p}\left(\alpha_{1},\dots,\alpha_{p^{n}}\right)$ is the splitting field, but what do elements of this field look like exactly?

I've factored $x^{p^{n}}-1$ as follows but I'm not sure what else to do $$\begin{align*} x^{p^{n}}-1 & =\left(x-1\right)\frac{x^{p^{n}}-1}{x-1}\\ & =\left(x-1\right)\prod_{i=0}^{n-1}\frac{x^{p^{i+1}}-1}{x^{p^{i}}-1}\\ & =\left(x-1\right)\prod_{i=0}^{n-1}\sum_{j=0}^{p-1}x^{p^{i}j} \end{align*}$$

Answer 1

I think you mean $x^{p^n}-x$. A field $F$ of characteristic $p$ satisfies $|F|=p^k$ for some integer $k$, so $|F^×|=|F \setminus \{0\}|$ $=p^k-1$. Thus as $p$ does not divide $p^k-1$ for any positive integer $k$, it follows by Lagrange's Thm that $|F^×|$ cannot have any group closed under multipication of order that is a multiple of $p$.

In a field $F$ of size $p^n$, all $p^n$ elements $\alpha \in F$ satisfy $\alpha^{p^n}-\alpha = 0$. So as the number of roots of $p(x)=x^{p^n}-x$ that are in $F$ is deg$(p)$, it follows that the polynomial $p(x)=x^{p^n}-x$ splits in $F$.

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Meanwhile in characteristic $p$ note that $(x-1)^{p^n}=x^{p^n}-1$. So $1$ is a root w multiplicity $p^n$; i.e., $x^{p^n}-1$ factors completely in $\mathbb{F}_p[x]$ to $(x-1)^{p^n}$. And thus the splitting field is simply the $p$-element field $\mathbb{F}_p$.

Answer 2

I’m assuming that $\Bbb Z_p$ means the ring of $p$-adic integers.

And if you really mean the polynomial $X^{p^n}-1$, i.e. if you’re really asking about the field of $p^n$-th roots of unity, then you may want to do the investigation in steps.

First, the $p$-th roots of unity, with minimal polynomial $X^{p-1}+X^{p-2}+\cdots+X^2+X+1$. By making the substitution $X=T+1$, you get a polynomial whose roots are of form $\zeta_p-1$, for primitive $p$-th roots of unity $\zeta_p$. You do the expansion and see that you have an Eisenstein polynomial of degree $p-1$, so this, your first level, is tamely ramified of degree $p-1$.

In succeeding degrees, the extensions are wild; each succeeding layer is (totally) wildly ramified of degree $p$. The whole extension thus is of degree $p^n-p^{n-1}$, with tame degree $p-1$ and wild degree $p^{n-1}$.

These extensions are important in the proof of the Local Kronecker-Weber Theorem. The global K-W Thm. says that the maximal abelian extension of $\Bbb Q$ is gotten by adjoining all roots of unity.