More Galois theory questions?

Question

* For two complex numbers a and b,
if a+b and a*b are algebraic over Q (the rationals), then why are a and b (separately) also algebraic?

* Why does X^(p^4) - X factor Over Fp (the finite field with p members, p of course is primary) into irreducibles of degrees 4,2 and 1 only?

Answer 1

This is two questions. I only answer one.

For the first, let A=a+b and y=ab. Then a and b are both solutions of x^2-Ax+B=0, so are in degree 2 extensions of Q(A,B). Since A and B are assumed algebraic, this is a finite extension, so is algebraic.

Answer 2

The proof given by Mathematician is excellent.
I thought of another:
Since a+b is algebraic, so is (a+b)² = a²+2ab+b².
Since ab is algebraic we get a²+b² algebraic.
Thus (a-b)² = a²-2ab+b² is algebraic,
which gives a-b algebraic
So (a+ b)/2 and (a-b)/2 are algebraic, and their sum, a,
and difference, b, are both algebraic.