(adsbygoogle = window.adsbygoogle || []).push({}); 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 [tex]\mathbb{F}_{2}/(x^4 + x^2 + 1)[/tex] differ from that of [tex]\mathbb{F}_{2}/(x^4 + x + 1)[/tex]?

2. Relevant equations

3. The attempt at a solution

Are there 16 elements in each of these quotient rings? Since [tex]2^4 = 16[/tex]. 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

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# General confusion about quotient rings and fields

Can you offer guidance or do you also need help?

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