proof of Minkowski’s bound

The proof of Minkowski’s bound will rely on Minkowski’s lattice point theorem (, but we first need to establish some lemmas.

Lemma 1.

Let M be a real number and suppose that for every non-zero ideal a of the ring of integers OK there exists a non-zero xa with norm N(x)MN(a).

Then, every ideal classMathworldPlanetmath of OK has a representative a satisfying N(a)M.


Let [𝔟] be an ideal class represented by the ideal 𝔟. Choosing a non-zero x𝔟 then x𝔟-1 is an ideal of 𝒪K and, by the condition of the lemma, contains a non-zero y satisfying N(y)MN(x𝔟-1). Then, 𝔞x-1y𝔟 is an ideal representing [𝔟] and N(𝔞)=N(y)/N(x𝔟-1)M. ∎

If the real embeddings of K are denoted by σk:K (k=1,,r1) and the complex embeddings are τk:K together with their complex conjugates τ¯k (k=1,,r2), then we define


Also note that r1×r2 is isomorphicPlanetmathPlanetmathPlanetmathPlanetmath as a real vector space to r1+2r2=n given by the isomorphismPlanetmathPlanetmath


As f and j are linear maps (with respect to the field of rationals ), the combinationMathworldPlanetmath fj gives a -linear map from K to n. The image will be a lattice, and we can compute its volume.

Lemma 2.

If a is a non-zero ideal of OK, then Γ=fj(a) is a lattice in Rn ( Its fundamental mesh has volume


The proof of this is to be added. ∎

Lemma 3.

For any L>0, let S be the set in Rr1×Cr2 consisting of points (x1,,xr1,y1,yr2) satisfying


Then, f(S) has volume (2r1-r2πr2/n!)Ln.


The proof of this is to be added. ∎

Proof of Minkowski’s bound

For an ideal 𝔞 and any constant b>1, let L>0 be given by


Letting S be the set given in Lemma 3 and Γ=fj(𝔞), Lemmas 2 and 3 give vol(S)>2nvol(Γ). As S is convex and symmetric about the origin, Minkowski’s theorem tells us that there is a non-zero x𝔞 with fj(x)S.

As the geometric mean is always bounded above by the arithmetic mean, we get the inequality


where MK=(n!/nn)(4/π)r2. If we choose b such that bMK|DK|N(𝔞) is less than the smallest integer greater than MK|DK|N(𝔞), then this gives N(x)MK|DK|N(𝔞) and Minkowski’s bound follows from Lemma 1.

Title proof of Minkowski’s bound
Canonical name ProofOfMinkowskisBound
Date of creation 2013-03-22 18:33:41
Last modified on 2013-03-22 18:33:41
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Proof
Classification msc 11R29
Classification msc 11H06
Related topic MinkowskisTheorem
Related topic MinkowskisConstant
Related topic IdealClass