# A finite field of $p^n$ elements exists for each prime power

I'm studying Fraleigh's Abstract Algebra, and I'm completely new to fields. I'm studying Theorem 33.10, which states:

A finite field of order $$p^n$$ exists for every prime power $$p^n.$$

The proof goes by considering $$\mathbb{Z}_p$$ and its algebraic closure $$\overline{\mathbb{Z}}_p$$. We let $$K\subseteq \overline{\mathbb{Z}}_p$$ denote the set of the distinct zeros of the polynomial $$x^{p^n} - x$$. Then, we show that $$K$$ is closed under addition, multiplication, has additive/multiplicative identities and inverses. From there we deduce that $$K$$ is a subfield of $$\overline{\mathbb{Z}}_p$$, with order $$p^n$$, as desired.

Well, here's what confuses me: Fraleigh states that $$K$$ is a subfield of $$\overline{\mathbb{Z}}_p$$ that contains $$\mathbb{Z}_p$$. I don't really understand why $$K$$ contains $$\mathbb{Z}_p$$. The reason must be fairly simple, because there's really no explanation, but please understand I am a complete beginner. Would appreciate some help.

• $x^p = x$ for all $x \in \mathbb{F}_p$. So $x^{p^2} = (x^p)^p = x^p = x$, and so on. Commented Jul 14, 2022 at 16:01
• A subfield contains the unit $1$. Therefore it contains $1+1+\dots+1$ for any number of terms. These make up $\mathbb{Z}_p$. Commented Jul 14, 2022 at 16:01
• @orangeskid Oh, so the elements of $\mathbb{Z}_p$ are all zeros of the polynomial? Wow, I should've just checked the definition of $K$. Thanks.
– Ssay
Commented Jul 14, 2022 at 16:04
• @GEdgar Thank you very much Edgar.
– Ssay
Commented Jul 14, 2022 at 16:06

Take a finite field $$K$$ with $$p$$ elements, and $$f\in K[t]$$ irreducible of order $$n$$. Let $$m$$ the ideal generated by $$f$$. We know that the quotient $$K[t]/m$$ is a field with $$p^n$$ elements.
An example for $$5^3$$. Take $$K=\mathbb{Z}_{5}$$ and $$f=t^3+t+1$$
• You need to prove that an irreducible $f$ of degree $n$ exists.
• Hi Guillerminho77, thanks for the answer, but I think the existence of such an irreducible $f$ is proven after my theorem in the question is proven. Thanks anyways.