This means that the number of elements in the associated Galois group is 1.

See the article on Galois groups for further explanation and examples.

We can then find that this Galois group has a transposition.

No direct description is known for the absolute Galois group of the rational numbers.

However, for other solvable Galois groups, the form of the roots can be much more complex.

This Galois group has only two elements: and the identity on .

Thus its Galois group modulo 3 contains an element of order 5.

Nowadays they are still a fundamental tool to compute Galois groups.

The absolute Galois group of an algebraically closed field is trivial.

It is thus necessary to have a notion of Galois group for an infinite algebraic extension.

In certain situations, the Galois group acts on other mathematical objects, for example a group.

Not every profinite group occurs as an absolute Galois group of a field.

The following is the simplest case where the Galois group is not abelian.

The motivic Galois group has the surrounding representation theory.

The conjecture concerns the absolute Galois group of the rational number field .

The Parker vector can assist in the recognition of Galois groups.

The construction in the preceding section used these generators to establish a polynomial's Galois group.

Some profinite groups occur as the absolute Galois group of non-isomorphic fields.

Any field with procyclic absolute Galois group is weakly C.

We wish to describe the Galois group of this polynomial, again over the field of rational numbers.

In this case, the group that exchanges the two roots is the Galois group belonging to the equation.

In this case the form of the solutions depends on the structure of this Galois group.

All finite groups do occur as Galois groups.

Let K be a field whose absolute Galois group G is finite.

H is the inverse limit of the Galois groups H.