using Minkowski’s constant to find a class number


We will use the theorem of Minkowski (see the parent entry (http://planetmath.org/MinkowskisConstant)).

Theorem (Minkowski’s Theorem).

Let K be a number fieldMathworldPlanetmath and let DK be its discriminantPlanetmathPlanetmathPlanetmath. Let n=r1+2r2 be the degree of K over Q, where r1 and r2 are the number of real and complex embeddings, respectively. The class groupMathworldPlanetmath of K is denoted by Cl(K). In any ideal class CCl(K), there exists an ideal AC such that:

|𝐍(𝔄)|MK|DK|

where N(A) denotes the absolute norm of A and

MK=n!nn(4π)r2.
Example 1.

The discriminants of the quadratic fields K2=(2),K3=(3) and K13=(13) are DK2=8,DK3=12 and DK13=13 respectively. For all three n=2=r1 and r2=0. Therefore, the Minkowski’s constants are:

MKi=12|DKi|,i=2,3,13

so in the three cases:

MKi1213=1.802

Now, suppose that C is an arbitrary class in Cl(Ki). By the theorem, there exists an ideal 𝔄, representative of C, such that:

|𝐍(𝔄)|<1.802<2

and therefore 𝐍(𝔄)=1. Since the only ideal of norm one is the trivial ideal 𝒪Ki, which is principal, the class C is also the trivial class in Cl(Ki). Hence there is only one class in the class group, and the class numberMathworldPlanetmath is one for the three fields K2,K3 and K13.

Example 2.

Let K=(17). The discriminant is DK=17 and the Minkowski’s bound reads:

MK=1217=2.06

Suppose that C is an arbitrary class in Cl(K). By the theorem, there exists an ideal 𝔄, representative of C, such that:

|𝐍(𝔄)|<2.06

and therefore 𝐍(𝔄)=1 or 2. However,

2=-3+1723+172

so the ideal 2𝒪K is split in K and the prime idealsPlanetmathPlanetmath

(-3+172),(3+172)

are the only ones of norm 2. Since they are principal, the class C is the trivial class, and the class group Cl(K) is trivial. Hence, the class number of (17) is one.

Title using Minkowski’s constant to find a class number
Canonical name UsingMinkowskisConstantToFindAClassNumber
Date of creation 2013-03-22 15:05:38
Last modified on 2013-03-22 15:05:38
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Example
Classification msc 11H06
Classification msc 11R29
Related topic ClassNumbersAndDiscriminantsTopicsOnClassGroups