SCHWEITZER 1970 Problem 1

We have $2n+1$ elements in the commutative ring $R$ : $$\alpha,\alpha_1,...,\alpha_n,\varrho_1,...,\varrho_n .$$ Let us define the elements $$\sigma_k=k\alpha + \sum_{i=1}^n \alpha_i\varrho_i^k .$$ Prove that the ideal $(\sigma_0,\sigma_1,...,\sigma_k,...)$ can be finitely generated.

L. Redei