# Fields and irreducible polynomial of $p^n$ degree

Let $K$ be a field of $p$ elements.

Let $f(x) \in K [x]$ be an irreducible polynomial of degree $n$.

Prove that the field $K[x]/(f(x))$ has $p^n$ elements.

By given theorem, let $K$ be a field, $P(x)\in K[x]$ an irreducible polynomial. Then $\exists$ a field $F$ s.t. $P(x)$ has a root in $F$, then $K \subseteq F$.

Below are some notes that may can get me the proof.

Can anyone provide me help?

• By the way, is it supposed to say, there exists a field $F$... AND $K\subseteq F$? – LASV Dec 5 '13 at 0:41

Hint 1: You can write all the elements of $K[x]/(f(x))$ in the form $a_nx^{\deg(f)-1}+b^{\deg(f)-2}+...+a_0+(f(x))$ (Why?)
Hint 2: Consider $K[x]/((P(x))$