Disquisitiones Arithmeticae

Carl Friedrich Gauss’s 1801 book established a systematic treatment of number theory, developing congruences, quadratic forms, and the theory of constructible regular polygons.

Congruence (number theory): A relation stating that two integers have the same remainder when divided by a given positive integer. Gauss introduced the notation and systematic use of congruences that organize much of the book.
Chinese remainder theorem: A theorem giving a unique solution modulo a product for compatible systems of congruences with coprime moduli. It exemplifies how congruences let Gauss combine separate modular conditions into one.
Carl Friedrich Gauss: A German mathematician whose work shaped number theory, algebra, statistics, and astronomy. Gauss wrote and published the book at age twenty-four.
Fundamental theorem of arithmetic: Every integer greater than one factors uniquely into prime numbers, apart from the order of the factors. The book treats unique prime factorization as a foundation for arithmetic investigations.
Gauss's lemma: A result relating the least positive residues of multiples of an integer to its quadratic character modulo an odd prime. Gauss uses the lemma as a key step in proving quadratic reciprocity.
Prime number: A positive integer greater than one whose only positive divisors are one and itself. The book develops results about primes, including criteria for testing whether numbers divide one another.
Euler's criterion: For an odd prime p and p not dividing a, it characterizes whether a is a quadratic residue using a power modulo p. The criterion supplies a computational test for quadratic residuosity in the book’s framework.
Leonhard Euler: An eighteenth-century Swiss mathematician whose work transformed analysis, number theory, and mathematical notation. Euler’s results on residues and primes formed part of the arithmetic tradition Gauss reorganized.
Constructible polygon: A polygon that can be drawn exactly using an unmarked straightedge and compass in finitely many steps. Gauss derived criteria for constructing regular polygons and proved the constructibility of the regular seventeen-gon.
Fermat number: An integer of the form 2^(2^n) + 1, named after Pierre de Fermat. Gauss’s criterion determines when a Fermat number yields a constructible regular polygon.

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