For a finite Galois extension, the Galois group is the group of field automorphisms of that are trivial on (i.e., the bijections that preserve addition and multiplication and that send elements of to themselves). The importance of this group stems from the fundamental theorem of Galois theory, which constructs an explicit one-to-one correspondence between the set of subgroups of and the set of intermediate extensions of the extension . By means of this correspondence, group-theoretic properties translate into facts about fields. For example, if the Galois group of a Galois extension as above is not solvable (cannot be built from abelian groups), then the zeros of cannot be expressed in terms of addition, multiplication, and radicals, i.e., expressions involving . For example, the symmetric groups is not solvable for . Consequently, as can be shown, the zeros of the following polynomials are not expressible by sums, products, and radicals. For the latter polynomial, this fact is known as the Abel–Ruffini theorem: (and ), (where is regarded as a polynomial in , for some indeterminates , is any field, and ).
For a finite Galois extension, the Galois group is the group of field automorphisms of that are trivial on (i.e., the bijections that preserve addition and multiplication and that send elements of to themselves). The importance of this group stems from the fundamental theorem of Galois theory, which constructs an explicit one-to-one correspondence between the set of subgroups of and the set of intermediate extensions of the extension . By means of this correspondence, group-theoretic properties translate into facts about fields. For example, if the Galois group of a Galois extension as above is not solvable (cannot be built from abelian groups), then the zeros of cannot be expressed in terms of addition, multiplication, and radicals, i.e., expressions involving . For example, the symmetric groups is not solvable for . Consequently, as can be shown, the zeros of the following polynomials are not expressible by sums products and radicals. For the latter polynomial, this fact is known as the Abel–Ruffini theorem: (and ), (w,here is regarded as a polynomial in , for some indeterminates , is any field, and ).