Root of a quadratic polynomial with a unit leading coefficient
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form
x2 + bx + c = 0
with b and c (usual) integers. When algebraic integers are considered, the usual integers are often called rational integers.
Common examples of quadratic integers are the square roots of rational integers, such as √2, and the complex number i = √−1, which generates the Gaussian integers. Another common example is the non-real cubic root of unity −1 + √−3/2, which generates the Eisenstein integers.
Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations, and other questions related to integral quadratic forms. The study of rings of quadratic integers is basic for many questions of algebraic number theory.
number theory, quadraticintegers are a generalization of the usual integers to quadratic fields. Quadraticintegers are algebraic integers of degree two...
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks...
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field...
square-free integer different from 0 {\displaystyle 0} and 1 {\displaystyle 1} . If d > 0 {\displaystyle d>0} , the corresponding quadratic field is called...
ideals. The quadraticinteger rings are helpful to illustrate Euclidean domains. Quadraticintegers are generalizations of the Gaussian integers in which...
Gaussian integers do not have a total ordering that respects arithmetic. Gaussian integers are algebraic integers and form the simplest ring of quadratic integers...
quadratic form on a vector space. The study of quadratic forms, in particular the question of whether a given integer can be the value of a quadratic...
quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers...
unity) is a quadraticinteger. For n = 5, 10, none of the non-real roots of unity (which satisfy a quartic equation) is a quadraticinteger, but the sum...
rounding-to-integer functions. The reason this satisfies N(ρ) < N(β), while the analogous procedure fails for most other quadraticinteger rings, is as...
symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form x 2 ≡ a mod p {\displaystyle x^{2}\equiv...
{\frac {1}{2}}(1+{\sqrt {d}}\,)} respectively. See Quadraticinteger for more. The ring of integers of the field F = Q [ α ] {\displaystyle F=\mathbb {Q}...
Hence, any 0–1 integer program (in which all variables have to be either 0 or 1) can be formulated as a quadratically constrained quadratic program. Since...
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:...
decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors, in which...
martingales Quadratic reciprocity, a theorem from number theory Quadratic residue, an integer that is a square modulo n Quadratic sieve, a modern integer factorization...
in quadratic form. A quadratic form with integer coefficients is called an integral binary quadratic form, often abbreviated to binary quadratic form...
binary quadratic forms is isomorphic to the narrow class group of Q ( d ) {\displaystyle \mathbf {Q} ({\sqrt {d}}\,)} . For real quadraticinteger rings...
catastrophes. The positive definite quadraticinteger matrix representative of all odd numbers contains the set of seven integers: {1, 3, 5, 7, 11, 15, 33} where...
major use in the formula for roots of a quadratic equation; quadratic fields and rings of quadraticintegers, which are based on square roots, are important...
unknown, it is called "univariate". The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation...
polynomial, namely bx − a. Quadratic irrational numbers, irrational solutions of a quadratic polynomial ax2 + bx + c with integer coefficients a, b, and c...
real-valued function of an integer or natural number variable). Examples of quadratic growth include: Any quadratic polynomial. Certain integer sequences such as...
{\displaystyle d} is a square-free integer and K = Q ( d ) {\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)} is the corresponding quadratic field, then O K {\displaystyle...
a discriminant is equivalent to a unique square-free integer. By a theorem of Jacobi, a quadratic form over a field of characteristic different from 2...
Quadratic probing is an open addressing scheme in computer programming for resolving hash collisions in hash tables. Quadratic probing operates by taking...
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