> compute the Galois group of polynomial x^5+x+1

To compute the Galois group of a polynomial, we first determine if the polynomial is irreducible. If it is, we then determine its splitting field and compute the Galois group as the group of automorphisms of the splitting field that fix the base field (in this case, the rationals).

The polynomial given is f(x) = x^5 + x + 1. Since the polynomial has no rational roots (by the Rational Root Theorem) and it is a polynomial with integer coefficients, it is irreducible over the rationals.

Now, let's find the splitting field of f(x). This step can be challenging, and there isn't a general algorithm to find the splitting field of any polynomial. In general, the splitting field will be a field extension of the rationals containing all the roots of the polynomial. For some polynomials, this step can be done easily, but for others (like this one), it can be quite difficult.

Without more information about the roots of f(x), we cannot explicitly compute the Galois group. However, we can say that the Galois group of this polynomial is a subgroup of the symmetric group S_5 (since the polynomial is of degree 5). The order of the Galois group is a divisor of 5! = 120. Depending on the properties of the roots of the polynomial, the Galois group could be any of the subgroups of S_5 that satisfy these conditions.

In summary, we can't compute the Galois group of f(x) = x^5 + x + 1 explicitly without more information about its roots. However, we do know that it is a subgroup of the symmetric group S_5, and its order is a divisor of 120.