Advanced Programming Guide

124 • Chapter 3: Programming with Modules⎡ ⎤0 1 1⎣ 0 0 1 ⎦0 0 0> AdjacencyMatrix( g2 );⎡ ⎤0 1 1⎣ 0 0 1 ⎦0 0 0Quotient FieldsAs an example of generic programming, a generic quotient field (or fieldof fractions) construction algorithm is discussed.Mathematical Description Given an integral domain D, its quotientfield is (up to isomorphism) the unique field k, paired with a nonzero ringhomomorphism η : D −→ k, with the property that, for any nonzero ringhomomorphism ϕ : D −→ F , in which F is a field, there is a unique ringhomomorphism σ for which the diagramkσηFDϕcommutes. Because a nonzero ring homomorphism into a field must beinjective, this says that every field F that contains D as a subring mustalso contain an isomorphic copy of k.Concretely, the quotient field of an integral domain D can be thoughtof as the set of “reduced fractions” n/d, with n, d ∈ D. A formal constructioncan be produced by defining an equivalence relation on the setD × (D \ {0}), according to which two pairs (n1, d1) and (n2, d2) areequivalent only if,n1 · d2 = n2 · d1.A representative from each equivalence class is chosen to represent thefield element defined by that class. This understanding guides the computerrepresentation of the quotient field.