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Math Help - factor ring tables

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    factor ring tables

    Draw up an addition and a multiplication table for the factor ring 2Z/8Z.
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  2. #2
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    Quote Originally Posted by Ryan0710 View Post
    Draw up an addition and a multiplication table for the factor ring 2Z/8Z.
    Let G=2\mathbb{Z} and H=8\mathbb{Z}.
    Then G/H = \{ H,2+H,4+H,6+H\}.
    Let a=2+H and e=H.
    Then G/H = \{ e, a,2a,3a\}.
    Adding them is simple, for example, 2a+3a = 5a=a by using 4a=e.
    Multiplying is similar, 2a\cdot 3a = 6a=2a.

    With those facts you can form a table.
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  3. #3
    Super Member Matt Westwood's Avatar
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    The elements of 2Z are ..., -6, -4, -2, 0, 2, 4, 6, 8, ...

    The elements of 8Z are ..., -42, -16, -8, -0, 8, 16, 24, ...

    The elements of 2Z/8Z are 0, 2, 4, 6.

    You can consider these as being 2x the elements in Z_4, i.e. 2 x {0, 1, 2, 3} and do all the work on Modulo 4 arithmetic and multiply everything by 2 when you've finished, or you can directly use modulo 8 arithmetic and create the multiplication and addition tables based on the defined subring of that.

    As long as you know how to do modulo n arithmetic you're home.
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