| For a given number field K, this dissertation focuses on counting the number of extensions L/K of a fixed degree, specified Galois closure, and bounded discriminant.;We begin in Chapter 1 with a historical overview of counting number fields by discriminant and outline a number of prior results on this and related problems.;In Chapter 2, we prove an upper-bound asymptotic on the number NK,n(X;G) of extensions L/K having the Galois group of the Galois closure of L/K isomorphic to G, and such that the classical discriminant Nm K/Q (DL/K) is at most X. We then give a tabulation of explicit upper bounds for particular Galois groups.;In Chapter 3, we generalize the results of Chapter 2 to general representations by introducing a new counting metric called the rho-discriminant, and we then prove an analogue of the counting theorem from Chapter 2 in this setting.;We conclude with a discussion of questions left open for future work. The main techniques involved in establishing the results are from the geometry of numbers, polynomial invariant theory, integral-point counting on schemes, and representation theory of finite groups. |
