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Izv. RAN. Ser. Mat., 2012, Volume 76, Issue 6, Pages 95–106 (Mi izv7331)  

This article is cited in 1 scientific paper (total in 1 paper)

Splitting fields of finite groups

D. D. Kiselev

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We give a simpler proof of the Goldschmidt–Isaacs theorem in the case $p>2$ and find new sufficient conditions for the applicability of the theorem in the case $p=2$. We thus obtain a theorem giving an estimate for the Schur index of an arbitrary irreducible complex representation of a finite group over the field of rational numbers. The proof of this theorem shows that in practical applications there is no need to verify sufficient conditions for the applicability of the Goldschmidt–Isaacs theorem in the case $p=2$: they can automatically be assumed to hold. We also prove a theorem on the connection between the realizability of any complex representation over the field of rational numbers of a finite group of odd order of a special type and the possibility of constructing regular polygons with straightedge and compasses.

Keywords: finite group, representation of a finite group, Schur index.

DOI: https://doi.org/10.4213/im7331

Full text: PDF file (472 kB)
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English version:
Izvestiya: Mathematics, 2012, 76:6, 1163–1174

Bibliographic databases:

UDC: 512.547.2
MSC: 20C15, 12F10
Received: 01.03.2011
Revised: 18.05.2012

Citation: D. D. Kiselev, “Splitting fields of finite groups”, Izv. RAN. Ser. Mat., 76:6 (2012), 95–106; Izv. Math., 76:6 (2012), 1163–1174

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. D. D. Kiselev, “A bound for the Schur index of irreducible representations of finite groups”, Sb. Math., 204:8 (2013), 1152–1160  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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