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General confusion about quotient rings and fields

  1. Feb 19, 2009 #1
    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,
  2. jcsd
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