Application of Galois theory i have a question regarding roots of equation,
find all $a$,such that the cubic polynomial $x^3-bx+a=0$ has three integer roots,
how can you solve these by using galois theory,what does the reducible polynomials,splitting fields,field extensions have to do with these,
explain each of them,in detail,because it serves as a introduction to galois theory,
ok for eg take $b$ to be 3 and list all $a$ such that equation has three integer roots

Suppose the polynomial has three integer roots $r_1, r_2, r_3$. Then $(x - r_1)(x - r_2)(x - r_3) = x^3 - bx + a$, hence
$$r_1 + r_2 + r_3 = 0$$
$$r_1 r_2 + r_2 r_3 + r_3 r_1 = -b$$
$$r_1 r_2 r_3 = -a.$$
Squaring the first equation gives $r_1^2 + r_2^2 + r_3^2 = 2b$, which immediately tells you that for fixed $b$ there are only finitely many possibilities for the roots, and from here it's casework for any fixed $b$.
For example, for $b = 3$ we get $r_1^2 + r_2^2 + r_3^2 = 6$, which has solutions $(\pm 1, \pm 1, \pm 2)$ up to cyclic permutation, and of these solutions only $(-1, -1, 2)$ and $(1, 1, -2)$ add up to zero. Hence the possible values in this case are $a = \pm 2$. 
