units of real cubic fields with exactly one real embedding

Let K be a number fieldMathworldPlanetmath with [K:]=3 such that K has exactly one real embedding. Thus, r=1 and s=1. Let 𝒪K* denote the group of units of the ring of integersMathworldPlanetmath of K. By Dirichlet’s unit theorem, 𝒪K*μ(K)× since r+s-1=1. The only roots of unityMathworldPlanetmath in K are 1 and -1 because K. Thus, μ(K)={1,-1}. Therefore, there exists u𝒪K* with u>1, such that every element of 𝒪K* is of the form ±un for some n.

Let ρ>0 and 0<θ<π such that the conjugatesPlanetmathPlanetmath of u are ρeiθ and ρe-iθ. Since u is a unit, N(u)=±1. Thus, ±1=N(u)=u(ρeiθ)(ρe-iθ)=uρ2. Since u>0 and ρ2>0, it must be the case that uρ2=1. Thus, u=1ρ2. One can then deduce that discu=-4sin2θ(ρ3+1ρ3-2cosθ)2. Since the maximum value of the polynomialPlanetmathPlanetmath 4sin2θ(x-2cosθ)2-4x2 is at most 16, one can deduce that |discu|4(u3+1u3+4). Define d=|disc𝒪K|. Then d|discu|4(u3+1u3+4). Thus, u3d4-4-1u3. From this, one can obtain that u3d-16+d2-32d+1928. (Note that a higher lower bound on u3 is desirable, and the one stated here is much higher than that stated in Marcus.) Thus, u2(d-16+d2-32d+1928)23. Therefore, if an element x𝒪K* can be found such that 1<x<(d-16+d2-32d+1928)23, then x=u.

Following are some applications:

  • The above is most applicable for finding the fundamental unitMathworldPlanetmath of a ring of integers of a pure cubic field. For example, if K=(23), then d=108, and the lower bound on u2 is (23+10212)23, which is larger than 9. Note that (43+23+1)(23-1)=2-1=1. Since 1<43+23+1<9, it follows that 43+23+1 is the fundamental unit of 𝒪K.

  • The above can also be used for any number field K with [K:]=3 such that K has exactly one real embedding. Let σ be the real embedding. Then the above produces the fundamental unit u of σ(K). Thus, σ-1(u) is a fundamental unit of K.


  • 1 Marcus, Daniel A. Number Fields. New York: Springer-Verlag, 1977.
Title units of real cubic fields with exactly one real embedding
Canonical name UnitsOfRealCubicFieldsWithExactlyOneRealEmbedding
Date of creation 2013-03-22 16:02:25
Last modified on 2013-03-22 16:02:25
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 13
Author Wkbj79 (1863)
Entry type Application
Classification msc 11R27
Classification msc 11R16
Classification msc 11R04
Related topic NormAndTraceOfAlgebraicNumber