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.

Subsections

• Residue Fields
• Localization
• Completion
• Transcendental Extension

Given a ring R and elements a_1, ..., a_r of R, which generate a maximal ideal I of R, create the residue field K of the quotient ring R/I, together with a map sending an element of R to the corresponding element of K.

Given a ring R and a maximal ideal I of R, create the residue field K of the quotient ring R/I, together with a map sending an element of R to the corresponding element of K.

Given a ring R and elements a_1, ..., a_r 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.

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.

Given a ring R and elements a_1, ..., a_r 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.

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.

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[x_1, ..., x_n] of R. This is equivalent to PolynomialRing(R, n).