Sum of powers

Provide the most elegant proof to the following claim you can produce. Suppose z_1,\ldots,z_s and w_1,\ldots,w_r are nonzero complex numbers and for every positive integer n it is true that z_1^n +\cdots+z_s^n = w_1^n+ \cdots+w_r^n. then r=s and the w_i are a reordering of the z_j.

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Sum of powers

Galois correspondence via group algebras

Suppose G is a finite group of automorphisms of a field E. Form the ring E(G) which is the subring of the ring of endomorphisms of E generated by G and multiplication by elements of E. Note that if \lambda\in E then the composite g\cdot \mu_\lambda equals the composite \mu_{g(\lambda)} \cdot g. Show that the E-subalgebras of the group ring E(G) are those of the form E(H) for H a subgroup of G.

Note By an E-subalgebra we mean those subrings that contain all the multiplications by elements \lambda \in E.

Galois correspondence via group algebras