Basic Number Theory

The first goal of algebraic number theory is the generalization of the theorem on the unique representation of natural numbers as products of prime numbers to algebraic numbers. Gauss considered the ring \( \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] \) of all numbers of the form \( a + \sqrt {{ - 1b}} \) with \( a,\;b \in \mathbb{Z} \) and showed that \( \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] \) is a ring with unique factorization in prime elements (see §2.1). He introduced these numbers for the development of his theory of biquadratic residues. Another motivation for the study of the arithmetic of algebraic numbers comes from the theory of Tiophantine equations. For example, the quadratic form \( f({x_1},\;{x_2}) = x_1^2 - Dx_2^2 \) with \( D \in \mathbb{Z} \), \( \sqrt {{D\; \notin \;\mathbb{Z}}} \) can be written in the form \( \left( {{x_1} - \sqrt {{D{x_2}}} } \right)\left( {{x_1} + \sqrt {{D{x_2}}} } \right) \). Hence the question about the representation of integers by f(a 1, a 2) with \( {a_1},\;{a_2} \in \mathbb{Z} \) can be reformulated as a question of factorization of algebraic numbers of the form \( {a_1} + \sqrt {{D{a_2}}} \). These numbers form a module in the field \( \mathbb{Q}(\sqrt {D} ) \).