Galois theory

In mathematics, Galois theory is a branch of abstract algebra.

Solvable groups and solution by radicals

The notion of a solvable group in group theory allows us to determine whether or not a polynomial is solvable in the radicals, depending on whether or not its Galois group has the property of solvability. In essence, each field extension L/K corresponds to a factor group in a composition series of the Galois group. If a factor group in the composition series is cyclic of order n, then the corresponding field extension is a radical extension, and the elements of L can then be expressed using the nth root of some element of K.

Related Topics:
Solvable group - Group theory - Factor group - Cyclic

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

If all the factor groups in its composition series are cyclic, the Galois group is called solvable, and all of the elements of the corresponding field can be found by repeatedly taking roots, products, and sums of elements from the base field (usually Q).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

One of the great triumphs of Galois Theory was the proof that for every n > 4, there exist polynomials of degree n which are not solvable by radicals—the Abel-Ruffini theorem. This is due to the fact that for n > 4 the symmetric group Sn contains a simple, non-cyclic, normal subgroup.

Related Topics:
Abel-Ruffini theorem - Symmetric group - Simple - Normal subgroup

~ ~ ~ ~ ~ ~ ~ ~ ~ ~