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
Valuation ring corresponding to the discrete non-Archimedean valuation vp of ℚ
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
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.
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.
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.
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.
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
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
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
Computation of an existing field as a extension of another field
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.
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
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