Constructive Algebra in Functional Programming and Type Theory

This thesis considers abstract algebra from a constructive point of view. The central concept of study is coherent rings − algebraic structures in which it is possible to solve homogeneous systems of linear equations. Three different algebraic theories are considered; Bézout domains, Prüfer domains and polynomial rings. The first two of these are non-Noetherian analogues of classical notions. The polynomial rings are presented from a constructive point of view with a treatment of Gröbner bases. The goal of the thesis is to study the proofs that these theories are coherent and explore how the proofs can be implemented in functional programming and type theory.