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Math Help - direct product of rings

  1. #1
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    direct product of rings

    I am stuck on this question:

    Show that when is any field in which .

    I don't know how to prove that a field is isomorphic to direct product.
    What properties do I need to use to prove it?

    Can anyone kindly give me some instructions how to do this question.?

    Thanks alot
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    I am stuck on this question:

    Show that when is any field in which .

    I don't know how to prove that a field is isomorphic to direct product.
    What properties do I need to use to prove it?

    Can anyone kindly give me some instructions how to do this question.?
    The ring \mathbb{F}(x)/(x^2-1) consists of elements of the form a+bx (where a,b\in\mathbb{F}), with algebraic operations (a+bx) + (c+dx) = (a+b) + (c+d)x and (a+bx)(c+dx) = (ac+bd) + (ad+bc)x. You have to think of a map from \mathbb{F}(x)/(x^2-1) to \mathbb{F}\times\mathbb{F} that preserves these operations.

    The obvious map a+bx\mapsto(a,b) preserves addition but not multiplication (because (a,b)(c,d) = (ab,cd) \ne (ac+bd,ad+bc)). So you need to think of a different way of mapping \mathbb{F}(x)/(x^2-1) to \mathbb{F}\times\mathbb{F}. I can't say more than that without giving away the whole solution.
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