quadratic fieldとは意味・読み方・使い方

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意味・対訳二次体、二次体 (にじたい、英: quadratic field) は、有理数体上、2次の代数体のことである。

Weblio英和対訳辞書での「quadratic field」の意味

quadratic field

二次体

quadratic field

二次体

二次体 (にじたい、英: quadratic field) は、有理数体上、2次の代数体のことである。

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NON-REPUDIATION SIGNING SYSTEM BASED ON QUADRATIC FIELD例文帳に追加

二次体に基づく否認不可署名方式 - 特許庁

例文

A squaring computation of GF(qn-1) on a quadratic extension field GF(qn)=GF(qn-12) is conducted on GF(qn-1).例文帳に追加

ＧＦ（ｑ_n-1）の２次拡大体ＧＦ（ｑ_n）＝ＧＦ（ｑ_n-1^2）上の自乗演算をＧＦ（ｑ_n-1）上で行う。 - 特許庁

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Wiktionary英語版での「quadratic field」の意味

quadratic field

名詞

quadratic field (複数形 quadratic fields)

(algebraic number theory) 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, LNCS 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.
- 1990, Alan Baker, Transcendental Number Theory, Cambridge University Press, page 47,
  The foundations of the theory of binary quadratic forms, the forerunner of our modern theory of quadratic fields, were laid by Gauss in his famous Disquisitiones Arithmeticae.
- 2000, Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, page 223,
  In this chapter, we consider the simplest of all number fields that are different from $\mathbb {Q}$ , i.e. quadratic fields. Since $n=2=r_{1}+2r_{2}$ , the signature $(r_{1},r_{2})$ of a quadratic field $K$ is either $(2,0)$ , in which case we will speak of real quadratic fields, or $(0,1)$ , in which case we will speak of imaginary (または complex) quadratic fields. By Proposition 4.8.11 we know that imaginary quadratic fields are those of negative discriminant, and that real quadratic fields are those with positive discriminant.
- 2007, H. M. Stark, The Gauss Class-Number Problems, William Duke, Yuri Tschinkel (editors), Analytic Number Theory: A Tribute to Gauss and Dirichlet, American Mathematical Society, Clay Mathematics Institute, page 247,
  Since Dedekind's time, these conjectures have been phrased in the language of quadratic fields. […] 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 $k$ .

使用する際の注意点

An equivalent definition derives from the fact that the quadratic fields are exactly the sets $\mathbb {Q} ({\sqrt {d}})=\left\{a+b{\sqrt {d}}:a,b\in \mathbb {Q} \right\}$ $\mathbb {Q} ({\sqrt {d}})=\left\{a+b{\sqrt {d}}:a,b\in \mathbb {Q} \right\}$ , where $d$ $d$ is a nonzero squarefree integer called the discriminant.
- It suffices to consider only squarefree integer discriminants. In principle (かつ as is sometimes stated), the discriminant may be rational; but, since $\textstyle \mathbb {Q} ({\sqrt {c^{2}d}})=\mathbb {Q} ({\sqrt {d}})$ , 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 $\textstyle x=a+b{\sqrt {d}}$ $\textstyle x=a+b{\sqrt {d}}$ (regarding this as a quadratic formula).
- If $d$ is positive, each $\textstyle a+b{\sqrt {d}}$ is real and $\textstyle \mathbb {Q} ({\sqrt {d}})$ is called a real quadratic field.
- If $d$ is negative, each $\textstyle \ a+b{\sqrt {d}}$ is complex and $\textstyle \mathbb {Q} ({\sqrt {d}})$ is called a complex quadratic field (sometimes, imaginary quadratic field).