4 - Rings and Fields

Summary

Extending the "integer" concept to algebraic numbers suggests the more general algebraic concept of ring. Likewise the concept of rational number suggests the algebraic concept of field. In this chapter we look specifically at fields of algebraic numbers and how to define their "integers." This involves the study of polynomial rings and the corresponding concepts of "prime" polynomial and "congruence modulo a prime." Then we return to algebraic number fields and view them "relative to" their subfields, such as the fields of rational numbers. This is facilitated by ideas from linear algebra, such as basis and dimension.