A factorization of $x^7+x^3+x-1=0$ over rationals

Question

I want to solve

$$x^7+x^3+x-1=0.$$

I tried all possible factorization ways (over rational coefficients), e.g.

$$x^7+x^3+x-1=(x^3+ax^2+bx+c)(x^4+rx^3+sx^2+tx+u)$$

But, all possible cases didn't work. Could this equation be related to Galois Theory? Because the equation could not be factored.

Answer 1

Its Galois group is $S_7$, you can compute it in magma

P< x >:=PolynomialAlgebra(Rationals());
f:=x^7+x^3+x-1;
G:=GaloisGroup(f);
print G;

It implies that must be irreducible, so you cannot factorice it in rationals.