In
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
, and
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
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
:
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 ]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 ...
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
:
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