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Math Help - Isomorphism

  1. #1
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    Isomorphism

    I must find an isomorphism between \mathbb{Z}_5[x]/(x^2+1) and \mathbb{Z}_5 \times \mathbb{Z}_5.

    The problem is that multiplication is defined entirely differently in the two rings. Any attempt by me to define an iso between the two breaks down at multiplication.
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  2. #2
    hpe
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    Quote Originally Posted by Treadstone 71
    I must find an isomorphism between \mathbb{Z}_5[x]/(x^2+1) and \mathbb{Z}_5 \times \mathbb{Z}_5.

    The problem is that multiplication is defined entirely differently in the two rings. Any attempt by me to define an iso between the two breaks down at multiplication.
    There is no canonical multiplication on \mathbb{Z}_5 \times \mathbb{Z}_5, so you can use the bijection \Phi: ax+b \mapsto (a,b) to lift the multiplication on \mathbb{Z}_5[x]/(x^2+1) to \mathbb{Z}_5 \times \mathbb{Z}_5. For a different quadratic polynomial, you would get a different ring structure (a field for \mathbb{Z}_5[x]/(x^2+2)).
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  3. #3
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    I'm not entirely sure what you mean by "lift" the multiplication. Isn't multiplication on Z5XZ5 defined as (a,b)(c,d)=(ac,bd)?
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  4. #4
    hpe
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    Quote Originally Posted by Treadstone 71
    I'm not entirely sure what you mean by "lift" the multiplication. Isn't multiplication on Z5XZ5 defined as (a,b)(c,d)=(ac,bd)?
    No. Just like multiplication on RxR = C isn't defined by (a,b)(c,d) = (ac, bd) but by (a,c)(b,d) = (ac-bd, ad+bc).
    Define \Phi: Z_5[x]/(x^2+1) \to Z_5 \times Z_5 by \Phi(ax+b) = (a,b), where ax+b is the unique representative of degree less than 2 of an element in Z_5[x]/(x^2+1). This is a bijection. Then define
    (a,b) \cdot (c,d) = \Phi\left( \Phi^{-1}(a,b) \cdot \Phi^{-1}(c,d) \right), where the multiplication on the right is the one in Z_5[x]/(x^2+1). Then work out what this means in Z_5 \times Z_5. Compare this to the formula for complex multiplication, and think about an explanation.

    Hope this helps.
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