1. ## ring+conditon=field

Let A be a ring with $p^n$ elements with characteristic p , $p$ prime, $n\geq 1$, and $f\in\mathbb{Z}[X]$ a polynom of degree $n$ ireductibile over $\mathbb{Z}_p$.If $f$ have a solution $a \in A$ then A is field .

2. Originally Posted by petter

Let A be a ring with $p^n$ elements with characteristic p , $p$ prime, $n\geq 1$, and $f\in\mathbb{Z}[X]$ a polynom of degree $n$ ireductibile over $\mathbb{Z}_p$.If $f$ have a solution $a \in A$ then A is field .
i'll suppose that the ring is unitary. since $\text{char} A = p,$ the map $[k] \hookrightarrow k.1_A$ is an embedding of $\mathbb{Z}_p$ into $A.$ let $\overline{f(x)}$ be the image of $f(x)$ under the natural homomorphism $\mathbb{Z}[x] \longrightarrow \mathbb{Z}_p[x].$

now let $\overline{f(a)}=0,$ where $a \in A.$ define the map $\nu: \mathbb{F}_{p^n}= \frac{\mathbb{Z}_p[x]}{<\overline{f(x)}>} \longrightarrow A$ by: $\nu(\overline{g(x)})=g(a), \ \forall g(x) \in \mathbb{Z}_p[x].$ it's easily seen that $\nu$ is a ring monomorphism. thus $\nu$ is an isomorphism

because $|\mathbb{F}_{p^n}|=|A|=p^n. \ \ \Box$

3. Can u give me a elementary solution please ?

4. Originally Posted by petter

Can u give me a elementary solution please ?