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.
- 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¶
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