Disquisitiones Arithmeticae¶
Gauss, C. F. (1801). Disquisitiones Arithmeticae.
Cited by¶
2 citations across 2 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Commutativity
- Mathematics-origin — commutativity is a foundational axiom in 19- and 20-century abstract algebra (Abel's work on solvable polynomials gave rise to "abelian" as the synonym for commutative in group theory; Gauss (1801) in Disquisitiones Arithmeticae gave the systematic groundwork for modular arithmetic and the commutative congruence calculus that anchors the theory; Noether (1921) later formalized commutative ring theory)
This sourceIntroduces the congruence relation a ≡ b (mod n) and the systematic treatment of modular arithmetic — supports the claim that Gauss gave the groundwork for the commutative congruence calculus.
Supported in partVerified against the work's full text
Gauss's Section 1 defines congruence mod m and introduces the congruence sign; it does not mention commutativity, Abel's work or Noether's ring theory.
“Hoc signum propter magnam analogiam quae inter aequalitatem ... congruentiam inueuitur adoptauimus”
- Mathematics-origin — commutativity is a foundational axiom in 19- and 20-century abstract algebra (Abel's work on solvable polynomials gave rise to "abelian" as the synonym for commutative in group theory; Gauss (1801) in Disquisitiones Arithmeticae gave the systematic groundwork for modular arithmetic and the commutative congruence calculus that anchors the theory; Noether (1921) later formalized commutative ring theory)
- Equivalence Relation
- … quotient set $\mathbb{Z}/n\mathbb{Z}$ inherits a ring structure from $\mathbb{Z}$ (because the congruence-modulo-$n$ relation is a congruence with respect to addition and multiplication), and the resulting modular-arithmetic framework underpins number theory from Fermat's little theorem and Euler's theorem onwards.
This sourceIntroduces the congruence notation a ≡ b (mod n) (n | a−b) and develops modular arithmetic systematically as the arithmetic of the residue (equivalence) classes. Cited in prose (Broad Use, Formal Example, Notes) on modular arithmetic as the originating use of equivalence-relation reasoning; no FACT marker.
- … quotient set $\mathbb{Z}/n\mathbb{Z}$ inherits a ring structure from $\mathbb{Z}$ (because the congruence-modulo-$n$ relation is a congruence with respect to addition and multiplication), and the resulting modular-arithmetic framework underpins number theory from Fermat's little theorem and Euler's theorem onwards.
Verification¶
Does it exist? Not checked yet. This entry carries no identifier to resolve. It was extracted from the citation as written in the article, normalized, and deduplicated against the rest of the registry.
Does it back the claim? Read against the text for 1 of 2 citations: 1 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.
See how references were verified.
Registry ID ref:1799d34ee645 · see in the full table