1
$\begingroup$

I am stucked to prove :

A finite field GF($p^n$) of $p^n$ elements exists for every prime power $p^n$.

In my book Fraleigh P303 ,

we think $\overline{Z_p}$ be an algebraic closure of ${Z_p}$ and let $K$ be the subset of $\overline{Z_p}$ consisting of all zeros of $x^{p^n} -x$ in $\overline{Z_p}$.

Then it seeme like $K$ has a prime characteristic $p$, since we use lemma when a field of prime characteristic $p$.

I want to understand why $K$ has a prime characteristic.

Thank you

$\endgroup$
1
$\begingroup$

Well, $p\cdot 1=0$ in $\mathbb{Z}_p$ and hence also in $\overline{\mathbb{Z}_p}$, and hence also in $K$. More generally if $k$ is a field, then any subfield or field extension has the same characteristic.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.