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.
Another different approach that maybe you coul find easier is this constructive argument:
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 $f=t^3+t+1$