Algebraic Number Theory

Algebraic Number Theory. Number fields, rings of integers, ideal class groups, and ramification. The literature on algebraic number theory divides naturally along several axes: the foundational structures that organise the subject, the techniques that drive proofs and computations, the questions about classification or representation that animate current research, and the bridges to neighbouring areas of mathematics and science. The references below trace those axes through the canonical textbook treatments and recent technical contributions.

Foundations and canonical references

The standard treatments of algebraic number theory approach the subject from complementary angles. Neukirch, Algebraic Number Theory (1999) is the anchor reference for the subject and lays out the core definitions, theorems, and worked examples that practitioners return to. Lang, Algebraic Number Theory (1994) gives a parallel, more proof-oriented exposition of the same material and is widely used as a graduate text. Cohen, A Course in Computational Algebraic Number Theory (1993) offers an alternative presentation that complements the primary references and is useful for triangulating definitions and proof techniques.

Open methodological questions for algebraic number theory include sharpening the bridges between foundational theory and computational practice, extending classical results to broader or more structured settings, and integrating the techniques surveyed above with adjacent mathematical disciplines. The references listed in this page are the entry points that current work builds on.