# Thread: field of order p^2

1. ## field of order p^2

Hi again!
My problem is to show that a field of order $\displaystyle p^2$ exists for every prime p.

In an earlier problem I found that there were $\displaystyle p^2$monic quadratics in$\displaystyle Z_p[x]$, but I don't know if that's useful. I also showed that $\displaystyle (1/2)(p^2 + p)$ of those were factorable. Again, don't know if that's related at all but thought I'd throw it out there. Any ideas or theorems would be super helpful, thanks!

2. Originally Posted by kimberu
Hi again!
My problem is to show that a field of order $\displaystyle p^2$ exists for every prime p.

In an earlier problem I found that there were $\displaystyle p^2$monic quadratics in$\displaystyle Z_p[x]$, but I don't know if that's useful. I also showed that $\displaystyle (1/2)(p^2 + p)$ of those were factorable. Again, don't know if that's related at all but thought I'd throw it out there. Any ideas or theorems would be super helpful, thanks!

Ok, let $\displaystyle p(x)\in\mathbb{F}_p[x]:=\mathbb{Z}_p[x]$ be one of those monic quadratics which is also irreducible ( can you prove such a thing must always exist? Note that according to what

you said you can!)) , so that the ideal $\displaystyle I:=<p(x)>$ is maximal in $\displaystyle \mathbb{F}_p[x]$ (why?) $\displaystyle \iff$ the factor ring $\displaystyle \mathbb{F}_{p^2}:=\mathbb{F}_p[x]/I$ is a field (why?) .

Well, now just prove that $\displaystyle \left|\mathbb{F}_{p^2}\right|=p^2$ ...

Hint: every element in $\displaystyle \mathbb{F}_{p^2}$ as defined above has the form $\displaystyle f(x)+I\,,\,\,f(x)\in\mathbb{F}_p[x]$ , but using Euclides algorithm in $\displaystyle \mathbb{F}_p[x]$ show that any element in $\displaystyle \mathbb{F}_{p^2}$ has a

representative $\displaystyle f(x)+I$ with $\displaystyle \deg f<2$ and any two of them is a different element in the field.

Well, now just count up all the different constant and linear poilynomials in $\displaystyle \mathbb{F}_p[x]$ ...

Tonio