7 Local Arithmetic Fields

7.1 Discrete Valuation Rings

Valuation rings are available for the rational field and for rational function fields. For rational function fields, given an arbitrary monic irreducible polynomial p(x)∈K[x], the valuation ring is

Op(x) = { f(x)g(x) : f(x),g(x)∈K[x],p(x) ∤g(x)}.
Valuations corresponding both to an irreducible element and to are allowed.

  • Valuation ring corresponding to the discrete non-Archimedean valuation vp of

  • Valuation ring corresponding to the discrete non-Archimedean valuation vp of a rational function field

  • Valuation ring corresponding to the valuation v of a rational function field

  • Arithmetic

  • Euclidean norm, valuation

  • Greatest common divisor

7.2 The Real and Complex Fields

The real and complex fields are different from most structures in that exact computation in them is almost never possible.

  • Arithmetic

  • Square root, arithmetic-geometric mean

  • Continued fraction expansion of a real number

  • Constants: π, Euler's constant, Catalan's constant

  • Logarithm, dilogarithm, exponential

  • Trigonometric functions, hyperbolic functions and their inverses

  • Bernoulli numbers

  • Γ function, incomplete Γ function, complementary incomplete Γ function, logarithm of Γ function

  • J-Bessel function, K-Bessel function

  • U-confluent hypergeometric function

  • Logarithmic integral, exponential integral

  • Error function, complementary error function

  • Dedekind η function

  • Jacobi sine theta-function and its k-th derivative

  • Log derivative (ψ) function, i.e, Γ'(x)Γ(x)

  • Riemann-ζ function

  • Polylogarithm, Zagier's modifications of the polylogarithm

  • Weber's f-function, Weber's f2-function, j-invariant

  • Integer polynomial having a given real or complex number as an approximate root (Hastad, Lagarias and Schnorr LLL-method)

  • Roots of an exact polynomial to a specified precision (Schönhage splitting circle method)

  • Summation of a series (Euler-Wijngaarden method for alternating series)

  • Numerical integration of a function (Romberg-type methods)

The real and complex fields in Magma are based on the GMP, MPFR and MPC packages. Some of the transcendental functions as well as root finding is based on code developed for PARI by Henri Cohen and others.

7.3 Newton Polygons

  • Construction of a newton polygon: Compact, infinite or including the origin

  • Construction of newton polygons from different types of data: f∈k[x,y], f∈k < < x > > [y], f∈k[y] and some prime object, a finite set of points, a finite set of faces (weighted dual vectors)

  • Finding faces, vertices and slopes

  • If polygon is derived from a polynomial f, finding restrictions of f to faces

  • Locating a given point relative to a newton polygon

  • Giving the valuations of the roots of a polynomial (with respect to a prime if not implicit)

Newton polygons can be used with polynomials over series rings in order to find roots of the polynomial.

  • Walker's [Wa78] algorithm for computing Puiseux expansions

  • Duval's [Duv89] algorithm for computing Puiseux expansions

7.4 p-adic Rings and their Extensions

A p-adic ring arises as the completion of the ring of integers at a prime while a local field arises as the completion at a prime ideal of a number field. Magma supports both fixed and free precision models, allowing the user to trade an increase in speed for automated precision management.

7.4.1 Construction

  • Construction of a p-adic ring or field (via polynomials or as completions)

  • Unramified extension of a local ring or field

  • Totally ramified extension of a local ring or field

  • Ring of integers of a local field

  • Field of fractions of a local ring

  • Change precision of a ring, field or element

  • Computation of a splitting field of an integral polynomial over an p-adic ring.

  • Enumeration of extensions of a given degree.

A local ring is a finite degree extension of a p-adic ring and may be either ramified or unramified or both. Any arbitrary tower of extensions can be constructed, as long as each step is either ramified or unramified or both.

7.4.2 Arithmetic

  • Arithmetic operations

  • Valuation of an element

  • Norm and trace of an element

  • Logarithm, exponential of an element

  • Square root, n-th root of an element

  • Minimal polynomial of an element over the p-adic subring or field

  • Image of an element under a power of the Frobenius automorphism

  • Linear algebra over local rings and fields

7.4.3 Polynomial Factorization

  • Polynomial algebra over local rings and fields

  • Greatest common divisor of two polynomials

  • Hensel lifting of the factors of a polynomial

  • Hensel lifting of the roots of a polynomial

  • Test a polynomial for irreducibility

  • Roots of a polynomial over a local ring or field

  • Factorization of polynomials over local ring or field

7.4.4 Class field theory

  • Unit group and norm group

  • Defining equations for abelian extensions

7.5 General Local Fields

In addition to the p-adic rings and their (ramified and unramified) extensions Magma contains local fields which are defined as an extension of another local field by any irreducible polynomial.

  • Construction of local fields as an extension by any irreducible polynomial

  • Calculation of subfields

  • Isomorphisms to extensions of p-adic fields

  • Construction of elements and basic arithmetic with those elements

  • Homomorphisms from local fields

  • Automorphism group and subgroups and fixed fields of these groups

7.6 Galois Rings

Magma provides facilities for computing with Galois rings. The features are currently very basic, but advanced features will be available in the near future, including support for the creation of subrings and appropriate embeddings, allowing lattices of compatible embeddings, just as for finite fields.

Because of the valuation defined on them, Galois rings are Euclidean rings, so they may be used in Magma in any place where general Euclidean rings are valid. This includes many matrix and module functions, and the computation of Gröbner bases. Linear codes over Galois rings will also be supported in the near future.

Features:

  • Creation of a default Galois ring (using a default defining polynomial).

  • Creation of a Galois ring by a specified defining polynomial.

  • Basic structural operations and arithmetic.

  • Euclidean operations.

7.7 Power, Laurent and Puiseux Series Rings

Magma contains an extensive package for formal power series. The fact that we may only work with a finite number of terms, n say, of a power series, i.e., a truncated power series, is made precise by noting that we are working in the quotient ring R[[x]]/ < xn+1 > , for some n, rather than in the full ring R[[x]]. Provided this is kept in mind, calculations with elements of a power series ring (though not field) are always precise.

Given a field K, a field of Laurent series K((x)) is regarded as the localization of the power series ring K[[x]] at the ideal < 0 > . More simply, it is the field of fractions of K[[x]]. Since elements of such a field are infinite series, calculation is necessarily approximate.

A power series ring R[[x]] is regarded as the completion of the polynomial ring R[x] at the ideal < 0 > .

Puiseux series with arbitrary fractional exponents are also supported (since V2.4).

  • Arithmetic

  • Inversion of units

  • Derivative, integral

  • Square root, valuation

  • Exponentiation, composition, convolution, reversion

  • Power series expansions of transcendental functions

  • R[[x]]/ < xn+1 > as an algebra over R

  • Factorization of polynomials over series rings

  • Unramified and ramified extensions of series rings

7.8 Lazy Power Series Rings

These power series rings contain only series of infinite precision. All coefficients of such series are computable but only finitely many will be known.

  • Creation of rings and elements

  • Arithmetic of elements

  • Retrieval of coefficients

  • Printing some specified terms of a series

  • Simple predicates on series

  • Derivative, integral and evaluation of series