$a + b + c, \ ab + bc + ca, \ abc$ Prove a, b and c are real. I have found this problem in a book. If a, b and c are complex numbers, such that $|a| \neq |b| \neq |c| \neq |a|$ and $a + b + c, \ ab + bc + ca, \ abc$ are all real numbers, we must prove $a, \ b, \ c$ are also real numbers. We can get $(a + b + c) ^ 2 \in \ \mathbb {R}$, so $a ^ 2 + b ^ 2 + c ^ 2 \in \mathbb{R}$. Then, using $a ^ 3 + b ^ 3 + c ^ 3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab -bc-ac) =  3(a + b)(b + c)(a+c)$ I can get $a ^ 3 + b ^ 3 + c ^ 3 \in \mathbb{R}$ and $(a+b)(b+c)(a+c) \in \mathbb{R}$. But I cannot fit these to prove that a, b or is real.

By Viète's relations, $a,b,c$ are the roots of
$$x^3-(a+b+c)x^2+(ab+bc+ca)x-abc$$
All the coefficients of this polynomial are real. If the polynomial has complex roots, there must be one such pair of them related by conjugation, but conjugate roots have the same magnitude, which is disallowed by the question. Thus the polynomial has only real roots, i.e. $a,b,c$ are real.
