algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...

, Minkowski's bound gives an

upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an eleme ...

of the norm of ideals to be checked in order to determine the class number of a

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

''K''. It is named for the mathematician Hermann Minkowski.

Definition

Let ''D'' be the discriminant of the field, ''n'' be the degree of ''K'' over

\mathbb

, and

2 r_2 = n - r_1

be the number of

complex embedding In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfi ...

s where

r_1

is the number of

real embedding In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfie ...

s. Then every class in the

ideal class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...

of ''K'' contains an integral ideal of

norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...

not exceeding Minkowski's bound :

M_K = \sqrt \left(\frac\right)^ \frac \ .

Minkowski's constant for the field ''K'' is this bound ''M''_''K''.Pohst & Zassenhaus (1989) p.384

Properties

Since the number of integral ideals of given norm is finite, the finiteness of the class number is an immediate consequence, and further, the

is generated by the prime ideals of norm at most ''M''_''K''. Minkowski's bound may be used to derive a lower bound for the discriminant of a field ''K'' given ''n'', ''r''₁ and ''r''₂. Since an integral ideal has norm at least one, we have 1 ≤ ''M''_''K'', so that :

\sqrt \ge \left(\frac\right)^ \frac \ge \left(\frac\right)^ \frac \ .

For ''n'' at least 2, it is easy to show that the lower bound is greater than 1, so we obtain Minkowski's Theorem, that the discriminant of every number field, other than Q, is non-trivial. This implies that the field of rational numbers has no

unramified extension In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...

Proof

The result is a consequence of Minkowski's theorem.

References

* * *

External links

* {{Planetmath reference, urlname=UsingMinkowskisConstantToFindAClassNumber, title=Using Minkowski's Constant To Find A Class Number *Stevenhagen, Peter
''Number Rings''.The Minkowski Bound
at Secret Blogging Seminar Theorems in algebraic number theory Hermann Minkowski