see Newton's identity.
Consider the power sums p1= x1 + ... + xn, p2= x1^2 + ... + xn^2,..., x1^n +...+ xn^n. These are polynomials in the elementary symmetric functions which Artin denotes by a1,...,an they must therefore be expressible as polynomials in a1,...,an. do this explicitly for n=4. Can you find the formula for arbitrary n?
see Newton's identity.