Faster access than browser!
24 relations: Algebraic number field, Carl Friedrich Gauss, Coprime integers, Cubic reciprocity, Elliptic function, Euler's criterion, Finite field, Gaussian integer, Gotthold Eisenstein, Ideal norm, Legendre symbol, Leopold Kronecker, Number theory, Proofs of Fermat's little theorem, Proofs of quadratic reciprocity, Quadratic reciprocity, Quadratic residue, Quartic reciprocity, Ring of integers, Root of unity, Springer Science+Business Media, Transfer (group theory), Trigonometric functions, Zolotarev's lemma.
Algebraic number field
In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
New!!: Gauss's lemma (number theory) and Algebraic number field · See more »
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.
New!!: Gauss's lemma (number theory) and Carl Friedrich Gauss · See more »
Coprime integers
In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.
New!!: Gauss's lemma (number theory) and Coprime integers · See more »
Cubic reciprocity
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x3 ≡ p (mod q) is solvable if and only if x3 ≡ q (mod p) is solvable.
New!!: Gauss's lemma (number theory) and Cubic reciprocity · See more »
Elliptic function
In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions.
New!!: Gauss's lemma (number theory) and Elliptic function · See more »
Euler's criterion
In number theory Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime.
New!!: Gauss's lemma (number theory) and Euler's criterion · See more »
Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.
New!!: Gauss's lemma (number theory) and Finite field · See more »
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers.
New!!: Gauss's lemma (number theory) and Gaussian integer · See more »
Gotthold Eisenstein
Ferdinand Gotthold Max Eisenstein (16 April 1823 – 11 October 1852) was a German mathematician.
New!!: Gauss's lemma (number theory) and Gotthold Eisenstein · See more »
Ideal norm
In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension.
New!!: Gauss's lemma (number theory) and Ideal norm · See more »
Legendre symbol
No description.
New!!: Gauss's lemma (number theory) and Legendre symbol · See more »
Leopold Kronecker
Leopold Kronecker (7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic.
New!!: Gauss's lemma (number theory) and Leopold Kronecker · See more »
Number theory
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
New!!: Gauss's lemma (number theory) and Number theory · See more »
Proofs of Fermat's little theorem
This article collects together a variety of proofs of Fermat's little theorem, which states that for every prime number p and every integer a (see modular arithmetic).
New!!: Gauss's lemma (number theory) and Proofs of Fermat's little theorem · See more »
Proofs of quadratic reciprocity
In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusual number of proofs.
New!!: Gauss's lemma (number theory) and Proofs of quadratic reciprocity · See more »
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers.
New!!: Gauss's lemma (number theory) and Quadratic reciprocity · See more »
Quadratic residue
In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.
New!!: Gauss's lemma (number theory) and Quadratic residue · See more »
Quartic reciprocity
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x4 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x4 ≡ p (mod q) to that of x4 ≡ q (mod p).
New!!: Gauss's lemma (number theory) and Quartic reciprocity · See more »
Ring of integers
In mathematics, the ring of integers of an algebraic number field is the ring of all integral elements contained in.
New!!: Gauss's lemma (number theory) and Ring of integers · See more »
Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.
New!!: Gauss's lemma (number theory) and Root of unity · See more »
Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
New!!: Gauss's lemma (number theory) and Springer Science+Business Media · See more »
Transfer (group theory)
In the mathematical field of group theory, the transfer defines, given a group G and a subgroup of finite index H, a group homomorphism from G to the abelianization of H. It can be used in conjunction with the Sylow theorems to obtain certain numerical results on the existence of finite simple groups.
New!!: Gauss's lemma (number theory) and Transfer (group theory) · See more »
Trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.
New!!: Gauss's lemma (number theory) and Trigonometric functions · See more »
Zolotarev's lemma
In number theory, Zolotarev's lemma states that the Legendre symbol for an integer a modulo an odd prime number p, where p does not divide a, can be computed as the sign of a permutation: where ε denotes the signature of a permutation and πa is the permutation of the nonzero residue classes mod p induced by multiplication by a. For example, take a.
New!!: Gauss's lemma (number theory) and Zolotarev's lemma · See more »
Redirects here:
References
[1] https://en.wikipedia.org/wiki/Gauss's_lemma_(number_theory)
