If both z + w and zw are real, how to prove that either z = w (conjugate) or Im(z) = Im(w) = 0?

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.3: Divisibility
Problem 30E: Let be as described in the proof of Theorem. Give a specific example of a positive element of .
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If both z + w and zw are real, how to prove that either z = w (conjugate) or Im(z) = Im(w) = 0?
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