Other Ring Constructions

Magma allows the construction of residue fields, localization of rings, and completion of rings. These constructions really just create appropriate rings of different categories within Magma.

Contents

Residue Class Fields

ResidueClassField(I) : Rng -> Fld, Map
Given a maximal ideal I of a ring R, create the residue class field K of the quotient ring R/I, together with a map sending an element of R to the corresponding element of K.

Localization

loc< R | a1, ..., ar > : Rng, RngElt, ..., RngElt -> Rng, Map
Given a ring R and elements a1, ..., ar of R, which generate a prime ideal P of R, create the localization L of R at P, together with a map sending an element of R to the corresponding element of L.
Localization(R, P) : Rng, Rng -> Rng, Map
Given a ring R and a prime ideal P of R, create the localization L of R at P, together with a map sending an element of R to the corresponding element of L.

Completion

comp< R | a1, ..., ar > : Rng, RngElt, ..., RngElt -> Rng, Map
Given a ring R and elements a1, ..., ar of R, which generate a prime ideal or zero ideal P of R, create the completion C of R at P, together with a map sending an element of R to the corresponding element of C.
Completion(R, P) : Rng, Rng -> Rng, Map
Given a ring R and a prime ideal or zero ideal P of R, create the completion C of R at P, together with a map sending an element of R to the corresponding element of C.

Transcendental Extension

ext< R | > : Rng -> RngUPol
Given a ring R create the univariate transcendental extension R[x] of R. This is equivalent to PolynomialRing(R).
ext< R, n | > : Rng, RngIntElt -> RngMPol
Given a ring R and an integer n ≥1, create the multivariate transcendental extension R[x1, ..., xn] of R. This is equivalent to PolynomialRing(R, n).
V2.28, 13 July 2023