Introduction

An affine algebra in Magma is simply the quotient ring of a multivariate polynomial ring P = R[x_1, ..., x_n] 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[x_1, ..., x_n]/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.

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.