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    Ring proofs

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    a. Is R= {a+b(squareroot of 2): a,b element of Z} a domain?
    b. Using the fact that alpha= (1/2)(1 + (square root of -19)) is a root of ((x^2)- x + 5), prove that R={a + b(alpha) : a,b element of Z} is a domain.

    Z= integers

    2. Assume that (x-a) divides f(x) in R[x]. Prove that (x-a)^2 divides f(x) if and only if (x-a) divides f prime in R[x].

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    Solution Preview

    1. It is routine to check that (R, +, .) is a commutative ring with unity. Now suppose that a + b(sqrt 2), and

    c + d(sqrt 2) are nonzero elements of R such that

    (a + b (sqrt 2))(c + d (sqrt 2)) = 0; then

    (ac + 2bd) + (ad + bd) sqrt 2 = 0, hence

    ac + 2bd = 0 and ad + bc = 0.

    These equations imply

    a/b = - 2d/c, and a/b = - c/d;

    that is,

    -2d/c = -c/d

    so that ...

    Solution Summary

    The solution provides examples of proving a ring is a domain, and working with a proof of divisibility.