quadratic field

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quadratic field

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quadratic field (plural quadratic fields)

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

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

• number field

• complex quadratic field, imaginary quadratic field, real quadratic field

• quadratic integer

• binary quadratic form
• quadratic form

• Algebraic number field on Wikipedia.Wikipedia
• Binary quadratic form on Wikipedia.Wikipedia
  • Quadratic form on Wikipedia.Wikipedia
• Quadratic irrational number on Wikipedia.Wikipedia
• Ideal class group on Wikipedia.Wikipedia
  • Class number problem on Wikipedia.Wikipedia
• Quadratic field on Encyclopedia of Mathematics
• Quadratic Field on Wolfram MathWorld