Given positive integers m_1,...,m_n, is it possible to solve the following equation system over the field of complex numbers?

m_1*x_1+...+m_n*x_n=0

m_1*x_1^2+...+m_n*x_n^2=0

......

m_1*x_1^(n-1)+...+m_n*x_n^(n-1)=0

x_1*x_2*...*x_n=1.

When n=1,2,3,4 I can find a formula for the solutions. But I can not do it for n>=4. Also, when m_1=m_2=...=m_n=1, the solutions can be written as

(\zeta^{\sigma(1)},\cdots,\zeta^{\sigma(n)})

where \zeta is a n-th primitive root of unity and \sigma is an element of the symmetric group S_n.

For other concrete examples, the Mathematica numerical computation shows that the number of solutions to this equation system would be n!. Does anybody have an idea to prove it?

In general, if we do not assume m_1,...,m_n are positive integers, what is the condition on m_1,...,m_n such that this equation system has a solution?