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Thread: Tables for a given field

  1. #1
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    Tables for a given field

    Give addition and multiplication tables for the field Z3/<x^2 + x + 2>.

    I am unsure how to find the congruence classes.

    Any help would be appreciated.
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  2. #2
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    Quote Originally Posted by page929 View Post
    Give addition and multiplication tables for the field Z3/<x^2 + x + 2>.

    I am unsure how to find the congruence classes.

    Any help would be appreciated.
    I think you mean $\displaystyle \mathbb{Z}_3[x]/\langle x^2+x+2\rangle$. So the elements of the field are polynomials in x (with $\displaystyle \mathbb{Z}_3$ coefficients), but when you factor out the ideal generated by $\displaystyle x^2+x+2$ you are effectively equating $\displaystyle x^2+x+2$ with 0. Thus $\displaystyle x^2 = -x-2 = 2x+1$, and you can use that identity to eliminate all powers of $\displaystyle x$ higher than 1. Therefore the field contains nine elements:

    $\displaystyle 0,\ 1,\ 2,\ x,\ x+1,\ x+2,\ 2x, \ 2x+1,\ 2x+2.$

    To get the addition and multiplication tables, work out the sum and product of each pair of elements using ordinary mod 3 arithmetic, but then substitute $\displaystyle 2x+1$ for any occurrences of $\displaystyle x^2$.
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