units of quadratic fields

Dirichlet’s unit theorem gives all units of an algebraic number fieldMathworldPlanetmath (ϑ) in the unique form


where ζ is a primitive wth root of unityMathworldPlanetmath in (ϑ), the ηj’s are the fundamental unitsMathworldPlanetmath of (ϑ),  0nw-1,  kjj,  t=r+s-1.

  • The case of a real quadratic fieldMathworldPlanetmath (m), the square-freem>1:  r=2,  s=0,  t=r+s-1=1.  So we obtain


    because  ζ=-1  is the only real primitive root of unity (w=2).  Thus, every real quadratic field has infinitely many units and a unique fundamental unit η.

    Examples:  If  m=3,  then  η=2+3;  if  m=421,  then  η=444939+216854212.

  • The case of any imaginary quadratic field (ϑ); here  ϑ=m,  the square-free  m<0:  The conjugatesPlanetmathPlanetmath of ϑ are the pure imaginary numbers ±m, hence  r=0,  2s=2,  t=r+s-1=0.  Thus we see that all units are


    1) m=-1.  The field contains the primitive fourth root of unity, e.g. i, and therefore all units in the field (i) are in, where  n=0, 1, 2, 3.

    2) m=-3.  The field in question is a cyclotomic fieldMathworldPlanetmath (http://planetmath.org/CyclotomicExtension) containing the primitive third root of unity and also the primitive sixth root of unity, namely


    hence all units are  ε=(1+-32)n,  where  n=0, 1,, 5, or, equivalently,   ε=±(-1+-32)n,  where  n=0, 1, 2.

    3) m=-2,  m<-3.   The only roots of unity in the field are ±1; hence  ζ=-1,  w=2,  and the units of the field are simply  (-1)n, where  n=0, 1.

Title units of quadratic fields
Canonical name UnitsOfQuadraticFields
Date of creation 2013-03-22 14:16:30
Last modified on 2013-03-22 14:16:30
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 39
Author pahio (2872)
Entry type Application
Classification msc 11R04
Classification msc 11R27
Synonym quadratic unit
Related topic Unit
Related topic NumberField
Related topic ImaginaryQuadraticField
Related topic SomethingRelatedToFundamentalUnits