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Thread: The elements of a field

  1. May 31st 2008, 11:43 AM #1
    toxiccan
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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. May 31st 2008, 03:10 PM #2
    NonCommAlg
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    Quote Originally Posted by toxiccan View Post
    Hi

    How to list the elements of the field \frac{\mathbb{Z}_2[x]}{<x^2+x+1>}?
    let <x^2+x+1>=I. then \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 \frac{\mathbb{Z}_5[i]}{<1+i>} ?
    since in \mathbb{Z}_5[i]: \ 1=(3+2i)(1+i), we have <1+i>=\mathbb{Z}_5[i] and thus the ring has only one element, i.e. \bar{0}.
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