# The degree of a splitting field over $\mathbb{F}_p$ of non-reducible monic polynomial of degree $n$

Let $$f\in\mathbb{F}_p(x)$$ be a monic irreducible polynomial, denoting $$\deg(f)=n$$.

I wish to show (if it's true) that $$f(x)$$'s splitting field is $$\mathbb{F}_{p^n}$$.

I did some manual test for some polynomials till degree $$5$$, using the expansion field $$\mathbb{F}_p[x]/\langle f(x) \rangle$$ , and used Frobenius' automorphism to show that if $$a$$ is a root then $$a^p , a^{p^2},a^{p^3}...a^{p^n}$$ are unique roots. When $$|\mathbb{F}_p[x]/\langle f(x) \rangle| = p^n$$. Is there a way to show that this generally?

Other explanations will be appreciated, however I am not familiar with Galois theory.

• Read about the finite fields of q=p^n elements, for example at Wikipedia. Here a link "en.wikipedia.org/wiki/Finite_field#Existence_and_uniqueness". – DrinkingDonuts Jan 30 at 17:30

## 1 Answer

All right, after some reading as proposed in comments, I believe this is the answer, please let me know If I'm wrong:

let $$a$$ be a root of $$f(x)$$ in $$\mathbb{F}_p[x]/\langle f \rangle$$ .

So $$|\mathbb{F}_p[x]/\langle f \rangle| = p^n$$. Thus it is isomorphic to $$\mathbb{F}_{p^n}$$ in which $$x^{p^n}-x$$ is splitting.

Thus $$a$$ is a root of $$x^{p^n}-x\in \mathbb{F}_p[x]$$.

$$f$$ is irreducible thus $$f(x) | p(x)$$ over $$\mathbb{F}_p[x]$$ because $$f$$ divides any polynomial with $$a$$ as a root. $$x^{p^n}-x$$ is splitting in $$\mathbb{F}_p[x]/\langle f \rangle\cong\mathbb{F}_{p^n}$$ thus $$f$$ must split in $$\mathbb{F}_p[x]/\langle f \rangle$$.

There is not subfield in which $$f$$ splits in, because any field containing a roots of $$f$$ , $$\mathbb{F}_p(a)$$ degree is $$n$$ and thus include $$p^n$$ elements.