What is algebraic number theory?

A number field $K$ is a finite degree algebraic extension of the rational numbers $\mathbf {Q}$. The primitive element theorem from Galois theory asserts that every such extension can be represented as the set of all polynomials of degree at most $d = [K:\mathbf{Q}] = \dim_{\mathbf{Q}} K$ in a single algebraic number $\alpha$:

$$K = \mathbf{Q}(\alpha) = \left\{ \sum_{n=0}^{m} a_n \alpha^n : a_n\in\mathbf{Q}\right\}.$$

Here $\alpha$ is a root of a polynomial with coefficients in $\mathbf {Q}$.

Algebraic number theory involves using techniques from (mostly commutative) algebra and finite group theory to gain a deeper understanding of the arithmetic of number fields and related objects (e.g., functions fields, elliptic curves, etc.). The main objects that we study in this book are number fields, rings of integers of number fields, unit groups, ideal class groups, norms, traces, discriminants, prime ideals, Hilbert and other class fields and associated reciprocity laws, zeta and $L$-functions, and algorithms for computing each of the above.