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An affine algebra in Magma is simply the quotient ring
of a multivariate polynomial ring P = R[x1, ..., xn]
by an ideal J of P. Such rings arise commonly in commutative
algebra and algebraic geometry. They can also be viewed as
generalizations of number fields and algebraic function fields,
when R is a field.
The elements of affine algebras are simply multivariate polynomials
which are always kept reduced to normal form modulo the ideal J of
"relations". Practically all operations which are applicable to
multivariate polynomials are also applicable in Magma to elements
of affine algebras (when meaningful).
If the ideal J of relations defining an affine algebra A =
R[x1, ..., xn]/J is maximal and R is a field, then A is a
field and may be used with any algorithms in Magma which work over
fields. Factorization of polynomials over such affine algebras is also
supported (including fields of small characteristic, since V2.10).
If an affine algebra defined over a field has finite dimension
considered as a vector space over the coefficient field, extra special
operations are available on its elements.
Currently the base ring R may be a field or a Euclidean ring.
Further operations for affine algebras over Euclidean rings
will be supported in the future.
An affine algebra has type RngMPolRes and its elements type
RngMPolResElt.
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