mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Shintani's unit theorem introduced by is a refinement of

Dirichlet's unit theorem In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a posi ...

and states that a subgroup of finite index of the totally positive units 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 ...

has a fundamental domain given by a rational polyhedric cone in the

Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...

of the field .

References

* * *{{citation, mr=0633664, last=Shintani, first= Takuro , chapter=A remark on zeta functions of algebraic number fields, title= Automorphic forms, representation theory and arithmetic (Bombay, 1979), pages= 255–260 , series=Tata Inst. Fund. Res. Studies in Math., volume= 10, publisher= Tata Inst. Fundamental Res., place= Bombay, year= 1981, isbn=3-540-10697-9

External links

Mathematical pictures
by Paul Gunnells Theorems in algebraic number theory