UVM Theses and Dissertations
Format:
Print
Author:
Musty, Michael J.
Dept./Program:
Mathematics
Year:
2014
Degree:
MS
Abstract:
In this work we describe a method for computing the Iwasawa A-invariant of a number field with abelian Galois group. We then provide some explicit computations using the Magma Computational Algebra System (see [1]).
The classical study of Iwasawa invariants of number fields dates back to Iwasawa's original work in the 1950s. Given a number field K and a prime p, let Kn denote the n-th layer in the cyclotomic Zp-extension of K. In [10), Iwasawa proves an asymptotic formula for the exact power of p dividing the class number of Kn. The Iwasawa invariants are defined by this asymptotic formula.
A general description of the Iwasawa [lambda]-invariants remains elusive, and a great deal of work has been done to compute this invariant in the case where K is an imaginary quadratic field (see [4]). For this case, an algorithm for computing the [lambda]-invariant explicitly can be found in [13]. Yet for more complicated examples, no such implementation appears to exist. Even in the next simplest case, where K is a cyclotomic field, we find, after a thorough search through the literature, only one paper where these [lambda]-invariants are computed explicitly (see [2]).
Following the work in [4, 2, 8, 13], we provide an explicit method to obtain [lambda]-invariants for K a cyclotomic field. We then include examples from our implementation of this method in Magma [1] which, at present, does not possess any functionality of this kind.
The classical study of Iwasawa invariants of number fields dates back to Iwasawa's original work in the 1950s. Given a number field K and a prime p, let Kn denote the n-th layer in the cyclotomic Zp-extension of K. In [10), Iwasawa proves an asymptotic formula for the exact power of p dividing the class number of Kn. The Iwasawa invariants are defined by this asymptotic formula.
A general description of the Iwasawa [lambda]-invariants remains elusive, and a great deal of work has been done to compute this invariant in the case where K is an imaginary quadratic field (see [4]). For this case, an algorithm for computing the [lambda]-invariant explicitly can be found in [13]. Yet for more complicated examples, no such implementation appears to exist. Even in the next simplest case, where K is a cyclotomic field, we find, after a thorough search through the literature, only one paper where these [lambda]-invariants are computed explicitly (see [2]).
Following the work in [4, 2, 8, 13], we provide an explicit method to obtain [lambda]-invariants for K a cyclotomic field. We then include examples from our implementation of this method in Magma [1] which, at present, does not possess any functionality of this kind.