Book: Disquisitiones Arithmeticae

Overview
Disquisitiones Arithmeticae (1801) by Carl Friedrich Gauss is a foundational monograph that systematically builds the modern theory of numbers. Written in Latin, it collects, clarifies, and extends a vast range of results in arithmetic, presenting them with unprecedented rigor and unified notation. The book established many of the central concepts and methods that would shape algebraic number theory for centuries.

Scope and Structure
The text is organized into logical sections that move from elementary congruences to deep structural results about forms and cyclotomy. Early chapters develop the language of congruences, residues, and modular arithmetic; later chapters treat quadratic forms, reciprocity laws, and cyclotomic equations. Each section contains definitions, propositions, and careful proofs designed to reveal both specific results and general principles.

Key Results
Gauss systematically formulates congruences and residue classes and gives a clear treatment of quadratic reciprocity, offering multiple proofs and illuminating its consequences. The theory of binary quadratic forms is developed comprehensively: classification, equivalence, composition, reduction, and the determination of which integers are represented by given forms. Crucial arithmetic structures such as discriminants, class numbers, and genera are described and used to obtain finiteness and representation results. The book also treats cyclotomy and the arithmetic of roots of unity, yielding criteria that underlie the constructibility of regular polygons, notably the celebrated proof that the regular 17-gon is constructible by straightedge and compass.

Methods and Innovations
A defining feature is Gauss's insistence on rigorous proof and systematic organization. Notation for congruence and residues, careful definitions of equivalence of forms, and the introduction of invariants like the discriminant provided a toolkit that made otherwise disparate observations coherent. Algebraic ideas are used in an arithmetic setting: the implicit use of ring-like structures, factorization in Gaussian integers, and congruence classes foreshadow later abstractions. Gauss blends computational examples with structural theorems, allowing methods to be both concrete and generalizable.

Conceptual Advances
Gauss's treatment of composition of binary quadratic forms anticipates group structures and class groups, making explicit the way forms combine and how equivalence classes behave. The emphasis on invariants and equivalence relations crystallized a new way of thinking about arithmetic objects up to transformation. The exploration of cyclotomy and sums involving roots of unity led to early instances of what would later be formalized as characters and Gauss sums, tools central to analytic and algebraic number theory.

Impact and Legacy
Disquisitiones Arithmeticae created the framework for modern number theory and influenced generations of mathematicians, including Dirichlet, Kummer, Dedekind, and Hilbert. Many lines of subsequent development, ideal theory, class field theory, and algebraic number theory, find their roots in Gauss's definitions and problems. The book remains a touchstone for its clarity, depth, and the enduring mathematical structures it uncovered.

Why it still matters
The concepts and techniques introduced continue to be taught and built upon in contemporary research and courses, from elementary congruences to advanced algebraic theories. Beyond specific theorems, the work's methodological discipline, combining calculation, invariant thinking, and structural proof, set a standard for mathematical exposition. Disquisitiones Arithmeticae stands as both a monumental source of results and a model for how deep, abstract theory can be grounded in concrete arithmetic questions.