quadratic field

Noun

 * 1)  A number field that is an extension field of degree two over the rational numbers.
 * 2) * 1985, Erich Kaltofen, Heinrich Rolletschek, Arithmetic in Quadratic Fields with Unique Factorization, Bob F. Caviness (editor), EUROCAL '85: European Conference on Computer Algebra, Proceedings, Volume 2, Springer, 204, page 279,
 * In a quadratic field $$\mathbf{Q}(\sqrt{D}),$$ $$D$$ a squarefree integer, with class number 1 any algebraic integer can be decomposed uniquely into primes but for only 21 domains Euclidean algorithms are known.
 * 1) * 2007, H. M. Stark, The Gauss Class-Number Problems, William Duke, Yuri Tschinkel (editors), Analytic Number Theory: A Tribute to Gauss and Dirichlet,, , page 247,
 * Since Dedekind's time, these conjectures have been phrased in the language of quadratic fields.Throughout this paper, $$k = \Q(\sqrt d)$$ will be a quadratic field of discriminant $$d$$ and $$h(k)$$ or sometimes $$h(d)$$ will be the class-number of $$k$$.
 * 1) * 2007, H. M. Stark, The Gauss Class-Number Problems, William Duke, Yuri Tschinkel (editors), Analytic Number Theory: A Tribute to Gauss and Dirichlet,, , page 247,
 * Since Dedekind's time, these conjectures have been phrased in the language of quadratic fields.Throughout this paper, $$k = \Q(\sqrt d)$$ will be a quadratic field of discriminant $$d$$ and $$h(k)$$ or sometimes $$h(d)$$ will be the class-number of $$k$$.

Usage notes

 * An equivalent definition derives from the fact that the quadratic fields are exactly the sets $$\Q(\sqrt{d}) = \left\{a+b\sqrt{d}:a,b\in\Q\right\}$$, where $$d$$ is a nonzero squarefree integer called the.
 * It suffices to consider only squarefree integer discriminants. In principle (and as is sometimes stated), the discriminant may be rational; but, since $$\textstyle\Q(\sqrt{c^2 d}) = \Q(\sqrt{d})$$, any given rational discriminant $$\textstyle\frac m n$$ can be replaced by the integer $$\textstyle n^2\frac m n = mn$$.
 * The discriminant exactly corresponds to the discriminant (the expression inside the surd) of the equation $$\textstyle x=a+b\sqrt{d}$$ (regarding this as a quadratic formula).
 * If $$d$$ is positive, each $$\textstyle a+b\sqrt{d}$$ is real and $$\textstyle\Q(\sqrt{d})$$ is called a.
 * If $$d$$ is negative, each $$\textstyle\ a+b\sqrt{d}$$ is complex and $$\textstyle\Q(\sqrt{d})$$ is called a (sometimes, ).

Translations

 * German: quadratischer Zahlkörper