# 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.

## 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 $a_1, \ldots, a_r$ 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 $L$.

### `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 $a_1, \ldots, a_r$ 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 $C$.

### `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 $n \geq 1$, create the multivariate transcendental extension $R[x_1, \ldots, x_n]$ of $R$. This is equivalent to `PolynomialRing(R, n)`.