Proving the ring of formal power series over a finite field is integral domain.

Question

Yes so my question shown in the first image;

I would like to prove that the ring of formal power series over the finite field of order prime, s is an integral domain.

So if it is not clear the first image is the question in its entirety and page 1 of my working out and the 2nd image is also my working out.

If anyone could verify my work, or tell me if I have got it all wrong lol it would be welcome.

-nomad609

Answer 1

This question can be done much easier by proving the converse: $$ fg\neq0 \iff f\neq0 \land g\neq0\tag1 $$ where $f,g\in K[[x]]$, $K$ is finite field of order $p$.

Let $a_0$ be constant of $f$ and $b_0$ be constant of $g$. Clearly $a_0b_0$ is constant of $fg$ and $$ p|a_0b_0\iff p|a_0 \lor p|b_0\quad\text{or}\quad p\nmid a_0b_0\iff p\nmid a_0 \land p\nmid b_0 $$ This means $$ (a_0b_0\not\equiv0\mod p) \iff (a_0\not\equiv0\mod p)\land (b_0\not\equiv0 \mod p) $$ So $(1)$ follows.