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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。

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quadratic field

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quadratic field

出典:『Wiktionary』 (2025/10/23 20:21 UTC 版)

名詞

quadratic field (plural 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 Q ( D ) , {\displaystyle \mathbf {Q} ({\sqrt {D}}),} $\mathbf {Q} ({\sqrt {D}}),$ D {\displaystyle 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:
- 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 Q {\displaystyle \mathbb {Q} } $\mathbb {Q}$ , i.e. quadratic fields. Since n = 2 = r 1 + 2 r 2 {\displaystyle n=2=r_{1}+2r_{2}} $n=2=r_{1}+2r_{2}$ , the signature ( r 1 , r 2 ) {\displaystyle (r_{1},r_{2})} $(r_{1},r_{2})$ of a quadratic field K {\displaystyle K} $K$ is either ( 2 , 0 ) {\displaystyle (2,0)} $(2,0)$ , in which case we will speak of real quadratic fields, or ( 0 , 1 ) {\displaystyle (0,1)} $(0,1)$ , in which case we will speak of imaginary (or 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 = Q ( d ) {\displaystyle k=\mathbb {Q} ({\sqrt {d}})} $k=\mathbb {Q} ({\sqrt {d}})$ will be a quadratic field of discriminant d {\displaystyle d} $d$ and h ( k ) {\displaystyle h(k)} $h(k)$ or sometimes h ( d ) {\displaystyle h(d)} $h(d)$ will be the class-number of k {\displaystyle k} $k$ .