Chapter I. Transition to Modern Number Theory

Abstract

This chapter establishes Gauss's Law of Quadratic Reciprocity, the theory of binary quadratic forms, and Dirichlet's Theorem on primes in arithmetic progressions.

Section 1 outlines how the three topics of the chapter occurred in natural sequence and marked a transition as the subject of number theory developed a coherence and moved toward the kind of algebraic number theory that is studied today.

Section 2 establishes quadratic reciprocity, which is a reduction formula providing a rapid method for deciding solvability of congruences $x^2\equiv m\bmod p$ for the unknown $x$ when $p$ is prime.

Sections 3–5 develop the theory of binary quadratic forms $ax^2+bxy+cy^2$, where $a,b,c$ are integers. The basic tool is that of proper equivalence of two such forms, which occurs when the two forms are related by an invertible linear substitution with integer coefficients and determinant 1. The theorems establish the finiteness of the number of proper equivalence classes for given discriminant, conditions for the representability of primes by forms of a given discriminant, canonical representatives of the finitely many proper equivalence classes of a given discriminant, a group law for proper equivalence classes of forms of the same discriminant that respects representability of integers by the classes, and a theory of genera that takes into account inequivalent forms whose values cannot be distinguished by linear congruences.

Sections 6–7 digress to leap forward historically and interpret the group law for proper equivalence classes of binary quadratic forms in terms of an equivalence relation on the nonzero ideals in the ring of integers of an associated quadratic number field.

Sections 8–10 concern Dirichlet's Theorem on primes in arithmetic progressions. Section 8 discusses Euler's product formula for $\sum_{n=1}^{\infty}n^{-s}$ and shows how Euler was able to modify it to prove that there are infinitely many primes $4k+1$ and infinitely many primes $4k+3$. Section 9 develops Dirichlet series as a tool to be used in the generalization, and Section 10 contains the proof of Dirichlet's Theorem. Section 8 uses some elementary real analysis, and Sections 9–10 use both elementary real analysis and elementary complex analysis.