prime ideal decomposition in quadratic extensions of

Let K be a quadratic number field, i.e. K=(d) for some square-free integer d. The discriminantPlanetmathPlanetmathPlanetmath of the extension is

DK/={d, if d1mod 4,4d, if d2,3mod 4.

Let 𝒪K denote the ring of integersMathworldPlanetmath of K. We have:

𝒪K{1+d2, if d1mod 4,d, if d2,3mod 4.

Prime idealsMathworldPlanetmathPlanetmath of decompose as follows in 𝒪K:

Theorem 1.

Let pZ be a prime.

  1. 1.

    If pd (divides), then p𝒪K=(p,d)2;

  2. 2.

    If d is odd, then

    2𝒪K={(2,1+d)2, if d3mod 4,(2,1+d2)(2,1-d2), if d1mod 8,𝑝𝑟𝑖𝑚𝑒, if d5mod 8.
  3. 3.

    If p2, p does not divide d, then

    p𝒪K={(p,n+d)(p,n-d), if dn2modp,𝑝𝑟𝑖𝑚𝑒, if d is not a square modp.


Title prime ideal decomposition in quadratic extensions of
Canonical name PrimeIdealDecompositionInQuadraticExtensionsOfmathbbQ
Date of creation 2013-03-22 13:53:46
Last modified on 2013-03-22 13:53:46
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 7
Author alozano (2414)
Entry type Theorem
Classification msc 11R11
Related topic CalculatingTheSplittingOfPrimes
Related topic ExamplesOfPrimeIdealDecompositionInNumberFields
Related topic PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ
Related topic NumberField
Related topic SplittingAndRamificationInNumberFieldsAndGaloisExtensions