**Solvable Groups and Solution By Radicals**

The notion of a solvable group in group theory allows one to determine whether a polynomial is solvable in radicals, depending on whether 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*, and if in the corresponding field extension *L*/*K* the field *K* already contains a primitive *n*-th root of unity, then it is a radical extension and the elements of *L* can then be expressed using the *n*th root of some element of *K*.

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 radicalsâ€”the Abelâ€“Ruffini theorem. This is due to the fact that for *n* > 4 the symmetric group *S*_{n} contains a simple, non-cyclic, normal subgroup, namely *A*_{n}.

Read more about this topic: Galois Theory

### Famous quotes containing the words solvable, groups, solution and/or radicals:

“The problems of the world, AIDS, cancer, nuclear war, pollution, are, finally, no more *solvable* than the problem of a tree which has borne fruit: the apples are overripe and they are falling—what can be done?... Nothing can be done, and nothing needs to be done. Something is being done—the organism is preparing to rest.”

—David Mamet (b. 1947)

“... until both employers’ and workers’ *groups* assume responsibility for chastising their own recalcitrant children, they can vainly bay the moon about “ignorant” and “unfair” public criticism. Moreover, their failure to impose voluntarily upon their own *groups* codes of decency and honor will result in more and more necessity for government control.”

—Mary Barnett Gilson (1877–?)

“I herewith commission you to carry out all preparations with regard to ... a total *solution* of the Jewish question in those territories of Europe which are under German influence.... I furthermore charge you to submit to me as soon as possible a draft showing the ... measures already taken for the execution of the intended final *solution* of the Jewish question.”

—Hermann Goering (1893–1946)

“Generally young men are regarded as *radicals*. This is a popular misconception. The most conservative persons I ever met are college undergraduates. The *radicals* are the men past middle life.”

—Woodrow Wilson (1856–1924)