# General confusion about quotient rings and fields

1. Feb 19, 2009

### samp

1. The problem statement, all variables and given/known data
I'm having a hard time understanding quotient rings. I think an example would help me best understand them.

For example, how does the ring structure of $$\mathbb{F}_{2}/(x^4 + x^2 + 1)$$ differ from that of $$\mathbb{F}_{2}/(x^4 + x + 1)$$?

2. Relevant equations

3. The attempt at a solution
Are there 16 elements in each of these quotient rings? Since $$2^4 = 16$$. If so, why does the degree of the polynomial determine this? Are either of these fields? I think I need to figure out if the polynomials are irreducible, how do I do this? Do I find all irreducible elements of each quotient ring and try to divide the ideal-generating polynomials by them and see if I get remainder 0? If so, how do I know which elements are irreducible?

Sorry for my stupidity; any help at all would be really appreciated.
Thanks guys,
Sam