Effective Field Theory and Algebraic Geometry

Present fashion in field theory and algebraic geometry is to replace classical constructive proofs by elegant existence proofs. For example, it is rare for students to see an actual procedure for factoring polynomials in ℚ[X₁,…,X _n ] in the course of finding out that it is a unique factorization domain. But constructive factorization is the essential backbone of constructive demonstrations that every K-closed set is the union of finitely many K-varieties, (i.e. Hilbert’s basis theorem) and that every K-variety can be normalized.