6 Global Arithmetic Fields

6.1 Number Fields and their Orders

Number fields in Magma are (abstract) finite extensions of or other number fields that can be constructed in a large number of different ways to faciliate a multitude of internal and external applications. For example number fields can be defined by specifying a single irreducible polynomial, as subfields of other number fields, via Galois correspondence, from characters, from a range of geometric objects or simply as cyclotomic fields of a given order. Number fields and their orders support a huge number of algorithms, starting from simple arithmetic, the computation of integral closures (maximal orders) and class and unit group to sophisticated tools for Galois cohomology and Diophantine equations. As well as state of the art algorithms for general numebr fields, Magma also contains special methods for quadratic and cyclotomic fields.

Facilities for general number fields have been developed in a joint project with the KANT group in Berlin. Number fields interact easily with their completions thus allowing a modern, local-global approach to a large number of problems.

6.1.1 Number Fields

  • Arithmetic of elements

  • Construction of equation orders, maximal orders, arbitrary orders

  • Simple and relative extensions, extensions defined by several polynomials

  • Subfields

  • Transfer between relative and absolute representations

  • Discriminant, reduced discriminant, signature

  • Factorization of polynomials over number fields

  • Completions of absolute fields

  • Number fields with arbitrary bases

  • Computation of Hilbert class fields and general class fields

  • Representation of a number field as a vector space or algebra over a given coefficient field

  • Automorphism group, Galois group of the normal closure

  • Action of the automorphisms on ideals and ideal classes

  • Dirichlet and Hecke character group as duals of residue rings and ray class groups

  • Basic Galois-Cohomology, computations in H1 and H2

6.1.2 Orders and Fractional Ideals

  • Multiple relative extensions

  • Maximal order, integral basis (Round 2 and Round 4 algorithms for absolute fields, Round 2 for relative extensions and special methods for Kummer extensions)

  • Suborders, extension orders

  • Construction of integral and fractional ideals

  • Ideal arithmetic: product, quotient, gcd, lcm, colon ideal

  • Determination of whether an ideal is: integral, prime, principal

  • Decomposition of primes

  • Valuations of order elements and ideals at prime ideals

  • Completions at prime ideals

  • Factorization of an ideal

  • Residue field of an order modulo a prime ideal

  • Residue class ring of an order modulo an arbitrary ideal

  • Completion of absolute maximal orders at finite primes

6.1.3 Invariants

  • Class group: Conditional (GRH) and unconditional algorithms

  • Unit group: Conditional (GRH) and unconditional algorithms

  • S-Unit group for arbitrary (finite) set S of primes

  • Picard group (ring class group) for non-maximal orders

  • Regulator

  • Exceptional units

  • Ray class groups, unit group of ray class rings of absolute maximal orders.

  • p-Selmer groups

6.1.4 Diophantine (and other) Equations

  • Norm equations, relative norm equations (both in the field and the order case, testing for local solubility)

  • Simultaneous norm equations, splitting of 2-cocycles

  • Thue equations

  • Unit equations

  • Index form equations

  • Integral points on Mordell curves

  • Hilbert 90

6.1.5 Quadratic Fields

  • All functionality of number fields

  • Euclidean structure of ℚ(√d), for d = -1, – 2, – 3, – 7, – 11,2,3,5,13.

  • Class number (Shanks' algorithm)

  • Ideal class group (Buchmann's method)

  • Fundamental unit, conductor

  • Solution of norm equations (Cornaccia's algorithm for imaginary fields and Cremona's conics for real quadratic fields)

  • Facilities for binary quadratic forms (see Lattices and Quadratic Forms)

  • 2-class group using the Bosma–Stevenhagen method

6.1.6 Cyclotomic Fields

  • All functionality of number fields

  • Sparse representation for large fields

  • Conductor and cyclotomic order

  • Cyclotomic subfields

  • Creation of roots of unity

  • Minimization of elements into smaller fields

  • Conjugation and complex conjugation

6.2 Galois theory of Number fields

Magma contains a rich set of commands to use and analyze the Galois structure of number fields. Starting with Klüners's subfield algorithm that allows the computation of all subfields of extensions of without the knowlegde of the Galois group, a generalization by Fieker and Klüners of Staudhar's method to compute the Galois group of any rational polynomial (no degree restriction, reducible polynomials as well) to Klüners algorithms for automorphisms of abelian fields.

Magma has been able to compute Galois groups of irreducible polynomials of degree > 50 and of reducible polynomials of degree > 25.

  • Determination of subfields

  • Automorphism groups of normal and abelian fields

  • Action of automorphisms on ideals, class group

  • Isomorphisms and embeddings of number fields

  • Galois group of number fields (over or any absolute number field) with no degree restriction.

  • Galois correspondence: fixed fields, fixed groups

  • Solvability by radicals, virtual work in the splitting field

  • Ramification theory

  • Action on S-Units

The computation of the Galois group of the degree 32 polynomial f(f(f(f(x)))) for f := x2 – 2 takes about 5 seconds, to solve (by radicals) a polynomial x6 – 3x5 – 2x4 + 9x3 – x2 – 4x + 1 with Galois group 6T11 of order 48 takes also about 5 seconds.

6.3 Class Field Theory of Number Fields

Class field theory is one of the most important results in number theory of the 20th century. While thought to be totally theoretical for a long time, it is now very practical and used in a growing number of applications from tabulating fields to Diophantine equations and representation theory of finite groups.

The core of Magma's class field theory is Fieker's algorithm to compute defining equations for a class field that is parametrized as a quotient of a ray class group. While analytic contructions are also available (over imaginary quadratic fields), this algebraic method applies to all fields.

  • Ray class groups modulo any integral ideal

  • Computation of defining equations of class fields

  • Test if a class field is normal, central or abelian without computing defining equations

  • Special integral closure algorithms based on Kummer theory

  • Analytical construction of Hilbert Class polynomials over imaginary quadratic fields

  • Computation of Hilbert class fields and ring class fields

  • The norm group of an abelian number field can be computed

  • Norm symbols, Artin-map is available

  • Solvability of norm equations can be tested, extending the Hasse-Norm-Theorem

  • Extension of automorphisms of the base field

  • Galois cohomology: action on Ray class groups

  • Galois cohomology: decide if a 1 or 2-cocycle is trivial, explicit splitting of cocycles

  • Grunwald-Wang theorem: find a cyclic extension having prescribed degrees at a finite number of places.

  • Functionality with Hecke Grössencharacters in CM fields

6.4 General Algebraic Function Fields

A general algebraic function field F/k of n variables over a field k is a field extension F of k such that F is a field extension of finite degree of k(x1, .s, xn) for elements xi∈F which are algebraically independent over k.

6.4.1 Rational Function Fields

Given any field k and indeterminates x1,…, xn, the user may form the field of rational functions k(x1,…, xn) as the localization of the polynomial ring k[x1,…, xn] at the prime ideal < x1,…, xn > .

  • Creation of a rational function field of a given rank over a given ring

  • Retrieval of the ring of integers, coefficient ring and rank

  • Ring predicates

  • Arithmetic

  • Numerator and Denominator

  • Degree and weighted degree

  • Evaluation

  • Derivative

  • Partial fraction expansion

  • Partial fraction decomposition (squarefree or full factorization)

6.4.2 Algebraic Function Fields

Within Magma, algebraic function fields of one variable can be created by adjoining a root of an irreducible, separable polynomial in k(x)[y] to the rational function field k(x). If k is a finite field, the function field is said to be global. An algebraic function field can be extended to create fields of the form k(x,a1,…, ar) where each extension occurs by adjoining a root of an irreducible and separable polynomial. Extensions may be formed using several polynomials simultaneously giving a non simple representation.

  • Creation of simple, relative and non simple extensions and mixed towers thereof

  • Creation of extensions of the constant field using bivariate polynomials

  • Retrieval of information defining the field

  • Exact constant field and genus

  • Change of representation from finite degree extensions to infinite and vice versa

  • Change of coefficient field to one lower in the extension tower

  • Computation of subfields and automorphisms

  • Homomorphisms from function fields into any ring by specifying the image of the primitive element and an optional map on the coefficient field

  • Computation of Galois Groups of simple extensions of the rational function field with no degree restriction

  • k-Automorphisms, isomorphism testing and embeddings

  • L-polynomial and ζ-function

  • Construction of a function field with an extended constant field

  • Construction of Artin-Schreier-Witt extensions from finite dimensional Witt-vectors

  • Constructive class field theory using both algebraic and analytic (Drinfeld modules) methods

6.4.3 Orders of Algebraic Function Fields

  • Finite and infinite equation orders

  • Finite and infinite maximal orders using the Round 2 algorithm for extensions which are not Kummer or Artin–Schreier, a basis for a p-maximal order of a Kummer extension and a maximal order of an Artin–Schreier extension can be written down.

  • Creation of orders whose basis is a transformation of an existing order

  • Integral closure

  • Basis of the order with the option to have the elements returned in a specified ring

  • Simplification of an order to a transformation of its equation order

  • (S-)Unit Group and unit rank, independent and fundamental units and regulator

  • Ideal class group for maximal orders

  • Ring predicates

  • Basis size reduction for finite, simple, non relative orders

6.4.4 Elements of Algebraic Function Fields and their Orders

Elements of function fields and their orders have 3 different representations. These representations are implemented generally for function field elements and number field elements. Standard elements are represented using coefficients of the basis elements. Elements of orders (and fields) with a power basis are represented using a polynomial representation. Elements of all orders or fields may have a product representation, being thought of as a formal product of a list of elements each to the power of some exponent. This can be a great advantage when the element is prohibitively large when represented using coefficients.

  • Arithmetic and modular arithmetic

  • Predicates

  • Creation of random elements and conversion to and from sequences

  • Norm and trace with respect to any given coefficient ring

  • Representation matrix, minimal and characteristic polynomials

  • Numerators and Denominators with respect to a given order

  • Module generated by a sequence of elements

  • Strong approximation theorem

  • Power series expansion, mapping of elements into completions

6.4.5 Ideals of Orders of Algebraic Function Fields

  • Creation of ideals from generators or a basis

  • Arithmetic

  • Roots of ideals

  • Predicates for integrality, prime, principal zero and one ideals

  • Predicate for prime ideals determining the type of ramification

  • Intersection, GCD and LCM

  • Factorization

  • p-radicals and p-maximal orders

  • Taking valuations of elements and ideals at prime ideals

  • Denominator

  • Retrieving basis and generators

  • Residue class field and the map to and from the order into it

  • Completions

  • Ramification and inertia degree

6.4.6 Places of Algebraic Function Fields

  • Creation of places as zeros and poles of elements of a field

  • Creation from prime ideals

  • Creation of random places of global fields

  • Creation of places of a given degree of global fields

  • Decomposition of places

  • Arithmetic

  • Residue class field, lifting elements out of and evaluating functions into

  • Valuation of elements and expanding elements at a place

  • Completion of fields and orders at places of any degree

  • Ramification and inertia degree

  • Retrieval of generators and a uniformizing element

  • Weierstrass places

  • Counting the number of places of a given degree over the exact constant field of global fields

  • The Serre and Ihara bounds on the number of places of degree 1 over the exact constant field of global fields

6.4.7 Divisors of Algebraic Function Fields

  • Creation from places, elements and ideals

  • Canonical and different divisor

  • Arithmetic including GCD and LCM

  • Support and Degree

  • Numerator and Denominator

  • Testing for properties of effective, positive, principal, special and canonical

  • Riemann–Roch space ℒ(D) of a divisor D, given by a k-basis of algebraic functions

  • Reduction of a divisor

  • Index of Speciality

  • Gap numbers, ramification divisors, Wronskian orders and Weierstrass places

  • Parametrization of a field at a divisor

  • Number of smooth divisors of global fields

6.4.8 Differentials of Algebraic Function Fields

  • Creation of a differential space

  • Creation of differentials from field elements

  • Arithmetic

  • Valuation of a differential at a place

  • Divisor of a differential

  • Differential spaces and bases for given divisors

  • Space and basis of holomorphic differentials of a field

  • Differentiations of elements of a function field

  • Residue of a differential at a place of degree one

  • Cartier operator and representation matrix of the Cartier operator (global case)

  • Module generated by a sequence of differentials

6.4.9 Divisor Class Groups for Global Algebraic Function Fields

  • Bounds on the generation of the class group

  • Computation of the class number and approximations to it

  • Construction of the divisor class group, structure of the divisor class group, representation of divisor classes as abelian group elements

  • S-class group, S-units and S-regulator for a finite set of places S

  • Exact sequence

    0→U(S)→F×→Div(S)→Cl(S)→0

  • Image and preimage computation possible for the maps of the exact sequence

  • Similar functionality for the ideal class group of the finite maximal order

  • p-rank of the divisor class group (separate method) and Hasse–Witt invariant

  • Tate–Lichtenbaum pairing

  • Global units

6.4.10 Class Field Theory for Algebraic Function Fields

  • Ray divisor class groups

  • Defining equation for class fields

  • Conductor and norm group

  • Genus, discriminant, number of places of given degree

  • Decomposition type of places of the base field

  • Exact constant field

  • Drinfeld modules of rank 1, rings of twisted polynomials

The development of this module is a joint project with the KANT group.

6.5 Algebraically Closed Fields

Algebraically closed fields (ACF's) have the property that they always contain all the roots of any polynomial defined over them.

  • Construction of algebraic closures over a finite field, the rational field or a rational function field of any characteristic

  • Automatic extension of the field by the roots of any polynomial over the field, and operations on conjugates of roots

  • Basic arithmetic

  • All standard algorithms for rings over generic fields work over such fields

  • Minimal polynomial

  • Simplification of the field

  • Construction of the corresponding absolute field together with the isomorphism

  • Pruning of useless variables and relations

It is not possible to construct explicitly the closure of a field, but the system works by automatically constructing larger and larger algebraic extensions of an original base field as needed during a computation, thus giving the illusion of computing in the algebraic closure of the base field.

A similar system was suggested by D. Duval and others (the D5 system [Duval85]), but this has difficulty with the parallelism which occurs when one must compute with several conjugates of a root of a reducible polynomial, leading to situations where a certain expression evaluated at a root is invertible but evaluated at a conjugate of that root is not invertible.

The system developed for Magma by Allan Steel avoids these problems, and is described in [ACF1, ACF2]. Consequently, ACF's behave in the same way as any other field implemented in Magma; all standard algorithms implemented for generic fields and which use factorization work without change (for example, the Jordan form of a matrix).

The system avoids factorization over algebraic number fields when possible, and automatically splits the defining polynomials of a field when factors are found. The field may also be simplified and expressed as an absolute field. Especially significant is also the fact that all the Gröbner basis algorithms work well over ACF's. One can now compute the variety of any zero-dimensional multivariate polynomial ideal over the algebraic closure of its base field. Puiseux expansions of polynomials are now also computed using an algebraically closed field.

Since V2.13, one may construct the algebraic closure over a finite field or a rational function field of any characteristic. For rational function fields of very small characteristic, inseparable field extensions are handled properly (see [ACF2] for details).