Publisher Summary This chapter explains algebraic number fields and its discreteness, factoring polynomials, valuation theory, unit theorem, and finiteness of class group and their proofs. Number theory is a good test for constructive mathematics as it applies to both discrete and continuous constructions; the constructive development brings to light constructive difficulties that were not at all apparent. The ability to decide whether a polynomial is irreducible or it has a nonconstant factor is used repeatedly in classical expositions of algebraic number theory. The algebraic number theory appears prima facie constructive, and it is common for authors to give routines for the construction of the objects that occur in the subject. The chapter describes the early work by some of the researchers who gave a systematic constructive exposition of the algebraic number fields. However, the development using recursive function theory is also there and is in progress.