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In mathematics, a fundamental discriminant D is an integer invariant in the theory of integral binary quadratic forms. If Q(x, y) = ax2 + bxy + cy2 is a quadratic form with integer coefficients, then D = b2 − 4ac is the discriminant of Q(x, y). Conversely, every integer D with D ≡ 0, 1 (mod 4) is the discriminant of some binary quadratic form with integer coefficients. Thus, all such integers are referred to as discriminants in this theory. * D ≡ 1 (mod 4) and is square-free, * D = 4m, where m ≡ 2 or 3 (mod 4) and m is square-free. The first ten positive fundamental discriminants are:

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rdfs:label	Discriminante fundamental (es) Fundamental discriminant (en) Discriminante fundamental (pt) Фундаментальный дискриминант (ru)
rdfs:comment	In mathematics, a fundamental discriminant D is an integer invariant in the theory of integral binary quadratic forms. If Q(x, y) = ax2 + bxy + cy2 is a quadratic form with integer coefficients, then D = b2 − 4ac is the discriminant of Q(x, y). Conversely, every integer D with D ≡ 0, 1 (mod 4) is the discriminant of some binary quadratic form with integer coefficients. Thus, all such integers are referred to as discriminants in this theory. * D ≡ 1 (mod 4) and is square-free, * D = 4m, where m ≡ 2 or 3 (mod 4) and m is square-free. The first ten positive fundamental discriminants are: (en) En matemáticas, un discriminante fundamental D es un número entero invariante en la teoría de formas cuadráticas enteras. Si Q(x, y) = ax2 + bxy + cy2 es una forma cuadrática con coeficientes enteros, entonces D = b2 − 4ac es el discriminante de Q (x, y). Por el contrario, cada entero D con D ≡ 0, 1 (mod 4) es el discriminante de alguna forma cuadrática binaria con coeficientes enteros. Por lo tanto, todos estos enteros se denominan discriminantes en esta teoría. * D ≡ 1 (mod 4) y no tiene cuadrados, * D = 4m, donde m ≡ 2 o 3 (mod 4) y m carece de cuadrados. (es) Em matemática, um discriminante fundamental D é um invariante inteiro na teoria das formas quadráticas binárias integrais. Se Q(x, y) = ax2 + bxy + cy2 for uma forma quadrática com coeficientes inteiros, então D = b2 − 4ac é o discriminante de Q(x, y). Por outro lado, todo inteiro D com D ≡ 0, 1 (mod 4) é o discriminante de alguma forma quadrática binária com coeficientes inteiros. Sendo assim, todos esses inteiros são referidos como discriminantes na teoria. * D ≡ 1 (mod 4) e livre de quadrados, * D = 4m, em que m ≡ 2 ou 3 (mod 4) e m é livre de quadrados. (pt) Фундаментальный дискриминант D — это целочисленный инвариант в теории целочисленных квадратичных форм от двух переменных (бинарных квадатичных форм). Если является квадратичной формой с целыми коэффициентами, то является дискриминантом формы Q(x, y). Существуют явные условия конгруэнтности, которые дают множество фундаментальных дискриминантов. Конкретно — D является фундаментальным дискриминантом тогда и только тогда, когда выполняются следующие условия * D ≡ 1 (mod 4) и свободно от квадратов, * D = 4m, где m ≡ 2 или 3 (mod 4) и m и свободно от квадратов. (ru)
dcterms:subject	Algebraic number theory
Wikipage page ID	13799075 (xsd:integer)
Wikipage revision ID	1124168767 (xsd:integer)
Link from a Wikipage to another Wikipage	5 (number) 8 (number) Quadratic field Quadratic integer Invariant (mathematics) 12 (number) 13 (number) 17 (number) Mathematics Fundamental theorem of arithmetic Modular arithmetic 1 21 (number) 24 (number) 28 (number) 29 (number) 33 (number) Discriminant of an algebraic number field Isomorphism Quadratic form Prime number Algebraic number theory Binary quadratic form Discriminant Square-free integer If and only if Integer Pairwise relatively prime Springer-Verlag
Link from a Wikipage to an external page	https://archive.org/details/binaryquadraticf0000buel https://archive.org/details/binaryquadraticf0000buel/page/69
sameAs	Fundamental discriminant Fundamental discriminant Fundamental discriminant Fundamental discriminant Fundamental discriminant Fundamental discriminant
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:OEIS
has abstract	In mathematics, a fundamental discriminant D is an integer invariant in the theory of integral binary quadratic forms. If Q(x, y) = ax2 + bxy + cy2 is a quadratic form with integer coefficients, then D = b2 − 4ac is the discriminant of Q(x, y). Conversely, every integer D with D ≡ 0, 1 (mod 4) is the discriminant of some binary quadratic form with integer coefficients. Thus, all such integers are referred to as discriminants in this theory. There are explicit congruence conditions that give the set of fundamental discriminants. Specifically, D is a fundamental discriminant if and only if one of the following statements holds * D ≡ 1 (mod 4) and is square-free, * D = 4m, where m ≡ 2 or 3 (mod 4) and m is square-free. The first ten positive fundamental discriminants are: 1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 (sequence in the OEIS). The first ten negative fundamental discriminants are: −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 (sequence in the OEIS). (en) En matemáticas, un discriminante fundamental D es un número entero invariante en la teoría de formas cuadráticas enteras. Si Q(x, y) = ax2 + bxy + cy2 es una forma cuadrática con coeficientes enteros, entonces D = b2 − 4ac es el discriminante de Q (x, y). Por el contrario, cada entero D con D ≡ 0, 1 (mod 4) es el discriminante de alguna forma cuadrática binaria con coeficientes enteros. Por lo tanto, todos estos enteros se denominan discriminantes en esta teoría. Hay condiciones de congruencia explícitas que dan el conjunto de discriminantes fundamentales. Específicamente, D es un discriminante fundamental si, y solo si, una de las siguientes declaraciones es válida: * D ≡ 1 (mod 4) y no tiene cuadrados, * D = 4m, donde m ≡ 2 o 3 (mod 4) y m carece de cuadrados. Los primeros diez discriminantes fundamentales positivos son: 1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 (sucesión A003658 en OEIS). Los primeros diez discriminantes fundamentales negativos son: −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 (sucesión A003657 en OEIS). (es) Em matemática, um discriminante fundamental D é um invariante inteiro na teoria das formas quadráticas binárias integrais. Se Q(x, y) = ax2 + bxy + cy2 for uma forma quadrática com coeficientes inteiros, então D = b2 − 4ac é o discriminante de Q(x, y). Por outro lado, todo inteiro D com D ≡ 0, 1 (mod 4) é o discriminante de alguma forma quadrática binária com coeficientes inteiros. Sendo assim, todos esses inteiros são referidos como discriminantes na teoria. Existem condições de congruência explícitas que fornecem o conjunto de discriminantes fundamentais. Especificamente, D é um discriminante fundamental se, e somente se, uma das seguintes afirmações for verdadeira: * D ≡ 1 (mod 4) e livre de quadrados, * D = 4m, em que m ≡ 2 ou 3 (mod 4) e m é livre de quadrados. Os primeiros dez discriminantes fundamentais positivos são: 1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 (sequência na OEIS). Os primeiros dez discriminantes fundamentais negativos são: −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 (sequência na OEIS). (pt) Фундаментальный дискриминант D — это целочисленный инвариант в теории целочисленных квадратичных форм от двух переменных (бинарных квадатичных форм). Если является квадратичной формой с целыми коэффициентами, то является дискриминантом формы Q(x, y). Существуют явные условия конгруэнтности, которые дают множество фундаментальных дискриминантов. Конкретно — D является фундаментальным дискриминантом тогда и только тогда, когда выполняются следующие условия * D ≡ 1 (mod 4) и свободно от квадратов, * D = 4m, где m ≡ 2 или 3 (mod 4) и m и свободно от квадратов. Первые десять положительных фундаментальных дискриминантов: 1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 (последовательность в OEIS). Первые десять отрицательных фундаментальных дискриминантов: −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 (последовательность в OEIS). (ru)
gold:hypernym	Invariant
prov:wasDerivedFrom	wikipedia-en:Fundamental_discriminant?oldid=1124168767&ns=0
page length (characters) of wiki page	3714 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Fundamental_discriminant
is rdfs:seeAlso of	Discriminant
is Link from a Wikipage to another Wikipage of	Quadratic field Cubic field Class number formula Proof that 22/7 exceeds π Siegel zero Wieferich prime Dirichlet character Discriminant of an algebraic number field Binary quadratic form Srinivasa Ramanujan Stark–Heegner theorem Discriminant (disambiguation)
is foaf:primaryTopic of	wikipedia-en:Fundamental_discriminant

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