Kummer theory provides converse statements. When K contains n distinct nth roots of unity, it states that any cyclic extension of K of degree n is formed by extraction of an nth root. Further, if K× denotes the multiplicative group of non-zero elements of K, cyclic extensions of K of degree n correspond bijectively with cyclic subgroups of
that is, elements of K× modulo nth powers. The correspondence can be described explicitly as follows. Given a cyclic subgroup
the corresponding extension is given by
that is, by adjoining nth roots of elements of Δ to K. Conversely, if L is a Kummer extension of K, then Δ is recovered by the rule
In this case there is an isomorphism
given by
where α is any nth root of a in L.
Read more about Kummer Theory: Generalizations
Famous quotes containing the word theory:
“We have our little theory on all human and divine things. Poetry, the workings of genius itself, which, in all times, with one or another meaning, has been called Inspiration, and held to be mysterious and inscrutable, is no longer without its scientific exposition. The building of the lofty rhyme is like any other masonry or bricklaying: we have theories of its rise, height, decline and fall—which latter, it would seem, is now near, among all people.”
—Thomas Carlyle (1795–1881)