Report: Zahlbericht

Overview
Hilbert’s 1897 Zahlbericht presents a unified, axiomatic exposition of algebraic number theory that synthesizes and systematizes the work of Gauss, Kummer, Dedekind, Kronecker, and Minkowski. Framed around the concept of an algebraic number field and its ring of integers, it establishes a coherent language and toolkit, ideals, norms, traces, discriminants, the different, class groups, and zeta functions, that turned a patchwork of results into a structured theory. Its clarity and completeness made it the standard reference for decades and set the agenda for class field theory and the arithmetic of fields and extensions.

Foundations of number fields
The report begins with algebraic number fields K as finite extensions of the rationals and their rings of integers O_K. Hilbert develops the basic invariants: norm and trace from K to Q, integral bases, discriminants of bases and of fields, and the different ideal measuring ramification. He emphasizes structural properties of O_K as a Dedekind domain, where arithmetic can be studied via ideals rather than elements, thus overcoming failures of unique factorization of elements.

Ideal theory and class groups
Building on Dedekind, Hilbert systematizes fractional ideals, prime ideals, and unique factorization of ideals into primes. He defines the ideal class group as the quotient of fractional ideals by principal ideals and proves its finiteness, introducing the class number as a fundamental invariant of K. He analyzes how ideals behave under field extension, including the correspondence between ideals of O_K and O_L for L/K, the norm of ideals, and the structure of the class group in towers. Concepts such as the conductor, the different, and the relative discriminant provide a precise language for ramification and arithmetic comparison across fields.

Units, regulators, and the zeta function
The report gives a comprehensive account of Dirichlet’s unit theorem: the unit group O_K^× is finitely generated with rank r1 + r2 − 1, where r1 and r2 count real and complex embeddings. Hilbert introduces the regulator as a measure of the size of the unit lattice. He treats the Dedekind zeta function ζ_K(s), its Euler product, and its role in encoding prime ideal distribution. He relates the residue of ζ_K(s) at s = 1 to the class number, regulator, number of roots of unity, and the discriminant, presenting the class number formula as a central bridge between analytic and algebraic invariants.

Decomposition, ramification, and reciprocity
Hilbert develops a systematic theory of the decomposition of rational primes in O_K via factorization of minimal polynomials modulo p, introducing inertia degrees and ramification indices and linking them to discriminants and the different. He gives a unified view of cyclotomic fields and Kummer extensions, explaining how congruences govern splitting behavior. Reciprocity laws are reformulated in the language of norm residues, providing a conceptual framework that subsumes quadratic reciprocity and anticipates broader abelian reciprocity. A notable highlight is the statement now known as Hilbert’s Theorem 90 for cyclic extensions L/K, which characterizes elements of norm 1 as ratios y/σ(y) for a generator σ of Gal(L/K), a cornerstone for later cohomological treatments of class field theory.

Methods and legacy
Hilbert integrates analytic methods and Minkowski’s geometry of numbers with ideal-theoretic arguments to prove finiteness and structural theorems. He standardizes notation and emphasizes axiomatic clarity, turning isolated techniques into a general methodology. The Zahlbericht not only codifies the arithmetic of algebraic number fields but also points toward the local-global perspective and the program of describing abelian extensions by arithmetic invariants. Its synthesis and vision directly influenced Takagi’s class field theory, Artin’s reciprocity, and later adelic and idelic formulations, while its language of ideals, norms, discriminants, and regulators remains foundational in modern number theory.