How to list all irreducible polynomials in a field?

I am currently trying to refresh my memory on some basic primary polynomials, apologies if my terminologies aren't correct:

For example, I have a field $\Bbb F_{2^3}$ and generates a list of irreducible polynomails such as below $$1$$ $$x$$ $$x^2$$ $$\vdots$$ $$x^3+x^2+x+1 \ .$$

Using the example above, I have three questions:

1. Sometimes there are irreducible polynomials with cofficients such as $x^2+2$ or $2x^2+1$ , how do I go about generating the coefficients, the only way I can generate it is without coefficients

2. How do I convert something like $x^4$ into the field above?

3. Sometimes a field can be generated by the polynomial such as $1+x+x^2$. how are the irreducible polynomials generated if the field is generated by another polynomial? this question doesn't make sense

Thanks!

• oops there shouldn't be a superscript 3 in the F, ill fix that right now. – Bonk Apr 3 '13 at 18:33
• Also: what do you mean "a field is generated by a polynomial"? Do you mean a quotient ring of the ring of all polynomials over a field by the (maximal) ideal generated by an irreducible polynomial? – DonAntonio Apr 3 '13 at 18:34
• yep, I must have mixed up my vocab, yes i meant irreducible poly. and when a field is generated by a poly means field only consist of valid output of that poly. e.g. $F_{10^2}=x^2$ means elements like 3 won't be in there because it's not a product of x^2 in any case. – Bonk Apr 3 '13 at 18:38
• This may be useful to you: Artin-Schreier Theory – Islands May 23 '13 at 21:53