In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically the field of
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 ob ...
, a Minkowski space is a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
associated with an
algebraic 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 ...
.
If ''K'' is a number field of
degree ''d'' then there are ''d'' distinct embeddings of ''K'' into C. We let ''K''
C be the image of ''K'' in the product C
''d'', considered as equipped with the usual
Hermitian inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. If ''c'' denotes
complex conjugation
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
, let ''K''
R denote the subspace of ''K''
C fixed by ''c'', equipped with a
scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
. This is the Minkowski space of ''K''.
See also
*
Geometry of numbers
Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice (group), lattice in \mathbb R^n, and the study of these lattices provides fundam ...
Footnotes
References
* {{cite book , first=Jürgen , last=Neukirch , authorlink=Jürgen Neukirch , title=Algebraic Number Theory , volume=322 , series=Grundlehren der Mathematischen Wissenschaften , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, year=1999 , isbn=978-3-540-65399-8 , zbl=0956.11021 , mr=1697859
Algebraic number theory