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    Algebraic theory

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    Proofs (see attached)

    Question 1
    Let b and d be distinct nonzero real numbers and c any real number .Prove that
    { b,c +di } is a basis of C over R.

    Hint-For any r + si ∈ C, r +si= (r/b-cs/bd)b +s/d(c+di).Hence {b,c + di} spans C over R. Prove that it is also linearly independent over R.

    Question 2
    If a+bi ∈ C and b≠0 ,prove that C= R(a +bi).


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    Solution Preview

    Please see the attachment.
    The statement is problem #2 is wrong and I gave a counter example.

    Question #1
    For any , I want to find , such that . Then we have and ...

    Solution Summary

    This is two proofs regarding abstract algebra.