# Splitting field of $X^{p^n}-1$ over $\mathbb{F}_p$

We know from Moore's theorem and the construction of finite fields that $$\mathbb{F}_{p^n}$$ is the splitting field of $$X^{p^n}-X$$ over $$\mathbb{F}_p$$. I was wondering what the $$X^{p^n}-1$$ splitting field would be and I imagine this is related to cyclotomic extensions, but we haven't studied those yet in my fields and Galois theory course.

I thought about looking at $$X^{p^n}-X$$ as a factor of a polynomial of the form $$X^{p^m}-X$$, but I wanted to know the canonical way to proceed.

• In characteristic $p$ we have $X^{p^n} - 1 = (X - 1)^{p^n}$, so the splitting field is $\mathbb{F}_p$. Apr 4 at 14:44
• @DanielArreola This should be an answer, not a comment.
– Mark
Apr 4 at 14:57
• Hi @DanielArreola Write it as a simple short answer and I will take it as it for the question (green check). : )
– IAG
Apr 4 at 15:14

Theorem (Kummer) Let $$p$$ be a prime, and $$m,n$$ be positive integers. The highest power of $$p$$ that divides $$\binom{n+m}{n}$$ is the number of carries when adding $$n$$ and $$m$$ in base $$p$$.
Corollary. Let $$p$$ be a prime, $$n\geq 1$$, and let $$k$$ be an integer, $$1\lt k\lt p^n$$. Then $$p\mid \binom{p^n}{k}$$.
Proof. There is at least one carry when adding $$p^n-k$$ and $$k$$ in base $$p$$, since both $$p^n-k$$ and $$k$$ have fewer than $$k+1$$ digits in base $$p$$, but their sum has $$k+1$$ digits. $$\Box$$
Corollary. If $$F$$ has characteristic $$p$$, and $$a,b\in F$$, then $$(a+b)^{p^n} = a^{p^n}+b^{p^n}$$.
Corollary. In $$\mathbb{F}_{p^k}[x]$$, for all $$r\geq 1$$ we have $$x^{p^r}-1 = (x-1)^{p^r}$$. In particular, $$x^{p^r}-1$$ splits over any field of characteristic $$p$$.
So the splitting field over $$\mathbb{F}_p$$ is $$\mathbb{F}_p$$ itself.