a quadratic field is an algebraic number field of degree two over Q {\displaystyle \mathbf {Q} } , the rational numbers. Every such quadratic field is...

only if the two fields are equal. In this case, the extension is a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and...

{\displaystyle 4x^{2}+2xy-3y^{2}} is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex...

quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation...

variable (x) Quadratic field, an algebraic number field of degree two over the field of rational numbers Quadratic irrational or "quadratic surd", an irrational...

called discriminant: the discriminant of an algebraic number field; the discriminant of a quadratic form; and more generally, the discriminant of a form, of...

theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime...

In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree...

In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative)...

In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as a x 2 + b x + c = 0 , {\displaystyle...

object appear to pulse. quadratic field strength A method of computing the mean strength of a varying stellar magnetic field. It is determined by calculating...

"real" property Real closed field Real quadratic field This disambiguation page lists articles associated with the title Real field. If an internal link led...

rank 1 exactly when the number field is a real quadratic field, a complex cubic field, or a totally imaginary quartic field. When the unit group has rank...

In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred...

century, binary quadratic forms have given up their preeminence in algebraic number theory to quadratic and more general number fields, but advances specific...

problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields Q ( d ) {\displaystyle...

is a quadratic field, the rank is 1 if it is a real quadratic field, and 0 if an imaginary quadratic field. The theory for real quadratic fields is essentially...

concerns the class number h of a real quadratic field of discriminant d > 0. If the fundamental unit of the field is ε = t + u d 2 {\displaystyle \varepsilon...

be greater than 1). One of the basic examples of norms comes from quadratic field extensions Q ( a ) / Q {\displaystyle \mathbb {Q} ({\sqrt {a}})/\mathbb...

formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as put into something like a final form...

{\displaystyle 7} ; this sum represents the largest square-free integer over a quadratic field of class number two, where 163 is the largest such (Heegner) number...

the minimal polynomial for α = √2 is a(x) = x − √2. In general, for the quadratic extension given by a square-free d {\displaystyle d} , computing the minimal...

is a direct sum of one-dimensional modules. Consider an imaginary quadratic field K = Q ( − d ) , d ∈ Z , d > 0 {\textstyle K=\mathbb {Q} \left({\sqrt...

In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise it...

numbers is quadratically closed; more generally, any algebraically closed field is quadratically closed. The field of real numbers is not quadratically closed...

field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity...

unique quadratic field k whose discriminant is d (in the case d = 1, the subfield Q is sometimes considered as the "degenerate" quadratic field of discriminant...

Hilbert about 10 years before. If R is the ring of integers of a real quadratic field, then the Hilbert modular group SL2(R) acts on the product H×H of two...

Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks...

the Heegner theorem establishes the complete list of the quadratic imaginary number fields whose rings of integers are principal ideal domains. It solves...