The goal of local class field theory is to classify abelian Galois extensions of a local field K. Several definitions of local fields are in use. In this thesis, local fields, which will be defined explicitly in Section 2, are fields that are complete with respect to a discrete valuation and have a finite residue field. A prototypical first example is Qp, the completion of Q with respect to the...

It gives a class of p-Borel principal ideals of a polynomial algebra over a field K for which the graded Betti numbers do not depend on the characteristic of K and the Koszul homology modules have monomial cyclic basis. Also it shows that all principal p-Borel ideals have binomial cycle basis on Koszul homology modules.

Let R = k[x1, . . . , xn] be a polynomial ring over a field k. Let J = {j1, . . . , jt} be a subset of {1, . . . , n}, and let mJ ⊂ R denote the ideal (xj1 , . . . , xjt). Given subsets J1, . . . , Js of {1, . . . , n} and positive integers a1, . . . , as, we study ideals of the form I = m1 J1 ∩ · · · ∩ m as Js . These ideals arise naturally, for example, in the study of fat points, tetrahedral...

A complete determination of the prime ideals invariant under winding automorphisms in the generic 3 × 3 quantum matrix algebra Oq(M3(k)) is obtained. Explicit generating sets consisting of quantum minors are given for all of these primes, thus verifying a general conjecture in the 3 × 3 case. The result relies heavily on certain tensor product decompositions for winding-invariant prime ideals, ...