Global Fields
Global fields in Magma include number fields and global function fields over finite fields. Algorithms compute rings of integers, ideal class groups, unit groups, ray class groups, and class field towers.
Contributors
Global Arithmetic Fields
An implementation of the Montes algorithm and its applications to speeding up computations involving ideal arithmetic in number fields and function fields has been contributed by Jens-Dietrich Bauch.
Jean-François Biasse (Calgary) implemented a quadratic sieve for computing the class group of a quadratic field. He also developed a generalisation of the sieve for number fields of degree greater than 2.
Florian Heß (TU Berlin) has contributed a major package for determining all isomorphisms between a pair of algebraic function fields.
David Kohel (Singapore–NUS, Magma) has contributed to the machinery for binary quadratic forms and has implemented rings of Witt vectors.
Jürgen Klüners (Düsseldorf) and Sebastian Pauli (UNC Greensboro) have developed algorithms for computing the Picard group of non-maximal orders and for embedding the unit group of non-maximal orders into the unit group of the field.
The facilities for general number fields and global function fields in Magma are based on the KANT V4 package developed by Michael Pohst and collaborators, first at Düsseldorf and then at TU Berlin. This package provides extensive machinery for computing with maximal orders of number fields and their ideals, Galois groups and function fields. Particularly noteworthy are functions for computing the class and unit group, and for solving Diophantine equations.
Code for finding all decompositions of a rational function into compositions of rational functions has been contributed by Jonas Szutkoski (Univ. Federal do Rio Grande do Sul).
The fast algorithm of Wieb Bosma and Peter Stevenhagen for computing the 2-part of the ideal class group of a quadratic field has been implemented by Mark Watkins (Bristol).
A database of number fields having degrees 2 through 9 was included in Magma from V2.19 (2013) onwards. The database was constructed by the Pari (Bordeaux) and KANT (Berlin) groups.
Galois Groups
The Fieker–Klüners algorithm for finding the Galois group of a polynomial has been improved in a number of ways over the period 2014–2016 by Stephan Elsenhans (Paderborn).
Jürgen Klüners (Kassel) has made major contributions to the Galois theory machinery for function fields and number fields. In particular, he implemented functions for constructing the subfield lattice and automorphism group of a field and also the subfield lattice of the normal closure of a field. In joint work with Claus Fieker (Magma), Jürgen has recently developed a new method for determining the Galois group of a polynomial of arbitrary high degree.
Jürgen Klüners (Kassel) and Gunter Malle (Kassel) made available their extensive tables of polynomials realising all Galois groups over ℚ up to degree 15.
Katharina Geißler contributed to the Handbook's coverage of Galois groups.