Quadratic field

English

Noun

1. Template:Lb A number field that is an extension field of degree two over the rational numbers.
   • 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, Template:W 204, page 279,

     In a quadratic field $𝐐(\sqrt{D}),$ $D$ 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.

   • Template:Quote-book
   • Template:Quote-book
   • 2007, H. M. Stark, The Gauss Class-Number Problems, William Duke, Yuri Tschinkel (editors), Analytic Number Theory: A Tribute to Gauss and Dirichlet, Template:W, Template:W, page 247,

     Since Dedekind's time, these conjectures have been phrased in the language of quadratic fields.Template:...Throughout this paper, $k=\mathbb{Q}(\sqrt{d})$ 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 $\mathbb{Q}(\sqrt{d})=\{a+b\sqrt{d}:a,b\in \mathbb{Q}\}$, where $d$ is a nonzero squarefree integer called the Template:M.
  • It suffices to consider only squarefree integer discriminants. In principle (and as is sometimes stated), the discriminant may be rational; but, since $\mathbb{Q}(\sqrt{c^{2}d})=\mathbb{Q}(\sqrt{d})$, any given rational discriminant $\frac{m}{n}$ can be replaced by the integer $n^2\frac{m}{n}=mn$.
• The discriminant exactly corresponds to the discriminant (the expression inside the surd) of the equation $x=a+b\sqrt{d}$ (regarding this as a quadratic formula).
  • If $d$ is positive, each $a+b\sqrt{d}$ is real and $\mathbb{Q}(\sqrt{d})$ is called a Template:M.
  • If $d$ is negative, each $a+b\sqrt{d}$ is complex and $\mathbb{Q}(\sqrt{d})$ is called a Template:M (sometimes, Template:M).

Hypernyms

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Hyponyms

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Related terms

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Translations

• German: Template:T

See also

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Further reading

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• Quadratic field on Template:W
• Quadratic Field on Template:W