Let $a_1, \dots, a_n$ and $b_1, \dots, b_n$ complex numbers such that $s_i(a_1, \dots, a_n) = s_i(b_1, \dots, b_n)$, for $i=1, \dots, n$ ( where $s_i$ are the elementary symmetric polynomials). How can I conclude that $a_k = b_k$ for $k=1, \dots, n$ ? Thanks in advance !
I have no ideas how to do it. Induction does not seem appropriated here.
Construct a polynomial using the values of the elementary symmetric polynomials as coefficients, according to Vieta. The n roots of this polynomial are the unique set of values which generate your given elementary symmetric polynomials.