zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the field of definition of a complex reflection group. (Sur le corps de définition d’un groupe de réflexions complexe.) (French) Zbl 0946.20024
Let V be a finite-dimensional vector space over the complex number field , G be a complex reflection group regarded as a finite subgroup of GL(V). The inclusion GGL(V) is called the natural representation of G. The extension field of the rational number field generated by the values of the characters of the natural representation of G is called the field of definition of G. M. Benard [J. Algebra 38, 318-342 (1976; Zbl 0327.20004)] announced the following theorem: Let K be the field of definition of a complex reflection group G, then all the complex representations of G are rational over K. The proof by Benard had some errors. Another proof is given in this paper. First, the conclusion is proved for the infinite families of complex reflection groups. Then for the exceptional groups, the computer is used to do the calculations.
MSC:
20G20Linear algebraic groups over , ,
20F55Reflection groups; Coxeter groups
51F15Reflection groups, reflection geometries
20C15Ordinary representations and characters of groups
20G05Representation theory of linear algebraic groups
20H15Other geometric groups, including crystallographic groups