Putnam 1973 B5

(a) Let $z$ be a solution of the quadratic equation $$az^2 +bz+c=0$$ and let $n$ be a positive integer. Show that $z$ can be expressed as a rational function of $z^n , a,b,c.$ (b) Using (a) or by any other means, express $x$ as a rational function of $x^{3}$ and $x+\frac{1}{x}.$