Disquisitiones Arithmeticae - Contents

Contents

The book is divided into seven sections, which are:

Section I. Congruent Numbers in General

Section II. Congruences of the First Degree

Section III. Residues of Powers

Section IV. Congruences of the Second Degree

Section V. Forms and Indeterminate Equations of the Second Degree

Section VI. Various Applications of the Preceding Discussions

Section VII. Equations Defining Sections of a Circle.

Sections I to III are essentially a review of previous results, including Fermat's little theorem, Wilson's theorem and the existence of primitive roots. Although few of the results in these first sections are original, Gauss was the first mathematician to bring this material together and treat it in a systematic way. He was also the first mathematician to realize the importance of the property of unique factorization (sometimes called the fundamental theorem of arithmetic), which he states and proves explicitly.

From Section IV onwards, much of the work is original. Section IV itself develops a proof of quadratic reciprocity; Section V, which takes up over half of the book, is a comprehensive analysis of binary and ternary quadratic forms. Section VI includes two different primality tests. Finally, Section VII is an analysis of cyclotomic polynomials, which concludes by giving the criteria that determine which regular polygons are constructible i.e. can be constructed with a compass and unmarked straight edge alone.

Gauss started to write an eighth section on higher order congruences, but he did not complete this, and it was published separately after his death.

The Disquisitiones was one of the last mathematical works to be written in scholarly Latin (an English translation was not published until 1965).