Alright. I dont know if I should put this question in its own forum post, but now I just add it here, because it's related to the other questions.

As you might have noticed, I mentioned that it was a three-part problem.

I've been looking at the third part, but I'm getting stuck here as well.

**(iii) Show that is an irreducible polynomial in and that is a field with 243 elements. Let . Find an element such that in .**
Alright. I've looked at it a bit. First I'm looking at the irreducability of

. I have a proposition that says

*"The ideal is a maximal ideal if and only if f is irreducible. In this case the quotient ring is a field."*.

So I'm guessing that first point on the agenda is proving that

is a maximal idea. But when I look through the stuff on maximal ideals in my book, all I find that seems usefull is

*"An ideal is maximal if and only if is a field."* But I have absolutely no idea on how to prove that

is a field.

I'm guessing that I have to show, that if we assume there is an ideal

where

and

implies that

. So if we imagine an ideal

that contains

and elements not part of

. Then we have to prove that

. The only way I can think of, is by proving that

.

I just dont know how to do that. Any tips?