Let be a field. Let be monic, and let be an extension of containing the roots of and . Say and . Define and in . Prove that and are in .
Note that, for each , . Thus . For a moment thinking of the as indeterminates, is fixed by any permutation of the . In particular, each coefficient of is a symmetric polynomial in the . Because the are the roots of the polynomial , every symmetric polynomial in the is in . So .
Suppose none of the is zero. Now , and so . So . Again, this product is fixed by any permutation of the , so that each coefficient of is symmetric in the . Thus .
If we have , where none of the roots of is zero, then by the previous argument the polynomial is in . Certainly then .