Kummer Theory
Kummer Extensions
Let
be a subfield of that contains the th root of unity , where is a prime number, and let be a Galois extension of degree . Then is obtained by adjoining a th root. In other words, is generated over by an element , with in .
Quintic Equations
The most famous result in Galois theory is the Abel-Ruffini theorem, which states that there is no general solution in radicals to polynomial equations of degree five or higher.