# 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 âˆˆ 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 âˆˆ C and bâ‰ 0 ,prove that C= R(a +bi).

C-complex

R-reals

https://brainmass.com/math/basic-algebra/algebraic-theory-179977

#### Solution Preview

Please see the attachment.

The statement is problem #2 is wrong and I gave a counter example.

Question #1

Proof:

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

#### Solution Summary

This is two proofs regarding abstract algebra.

$2.49