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Thread: The elements of a field

  1. #1
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    The elements of a field

    Hi

    How to list the elements of the field Z2[x]/<x^2+x+1>?
    and how many elements are in Z5[i]/<1+i> ?
    thx for the help
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  2. #2
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    Quote Originally Posted by toxiccan View Post
    Hi

    How to list the elements of the field $\displaystyle \frac{\mathbb{Z}_2[x]}{<x^2+x+1>}$?
    let $\displaystyle <x^2+x+1>=I.$ then $\displaystyle \frac{\mathbb{Z}_2[x]}{I}=\{a+bx + I: \ a,b \in \mathbb{Z}_2 \} = \{I,1+I,x+I,1+x+I\} \simeq \mathbb{F}_4.$

    and how many elements are in $\displaystyle \frac{\mathbb{Z}_5[i]}{<1+i>}$ ?
    since in $\displaystyle \mathbb{Z}_5[i]: \ 1=(3+2i)(1+i),$ we have $\displaystyle <1+i>=\mathbb{Z}_5[i]$ and thus the ring has only one element, i.e. $\displaystyle \bar{0}.$
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