Galois theory: Kummer extensions

This lecture, part of a graduate course on Galois theory, focuses on Kummer extensions. The speaker begins by recalling the connection between solvability by radicals and solvable Galois groups, then poses the converse problem: given a cyclic Galois extension of order n, when is it radical? He introduces assumptions: the base field contains all n-th roots of unity, the characteristic does not divide n, and n is prime. Under these, he proves that the extension is obtained by adjoining an n-th root of an element. The proof uses linear algebra: viewing the Galois group action as a linear transformation, he shows it is diagonalizable, and an eigenvector with non-trivial eigenvalue yields the desired element. He illustrates with the example of the cubic x^3 + x^2 - 2x - 1, whose roots are 2cos(2π/7), etc., and derives an explicit expression using cube roots. He then discusses when two such extensions coincide, leading to a classification by the quotient K*/K*^n. Finally, he introduces the Kummer pairing, a bilinear map between the Galois group of the maximal Kummer extension and K*/K*^n, noting a subtlety involving Tate twists. The lecture concludes by mentioning that the characteristic p case will be treated next, leading to Artin-Schreier extensions.