An infinite series of Kronecker conjugate polynomials
Author:
Peter Müller
Journal:
Proc. Amer. Math. Soc. 125 (1997), 19331940
MSC (1991):
Primary 11C08, 11R09, 20B05; Secondary 11R32, 12E05, 12F10
MathSciNet review:
1396989
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a field of characteristic , a transcendental over , and be the absolute Galois group of . Then two nonconstant polynomials are said to be Kronecker conjugate if an element of fixes a root of if and only if it fixes a root of . If is a number field, and where is the ring of integers of , then and are Kronecker conjugate if and only if the value set equals modulo all but finitely many nonzero prime ideals of . In 1968 H. Davenport suggested the study of this latter arithmetic property. The main progress is due to M. Fried, who showed that under certain assumptions the polynomials and differ by a linear substitution. Further, he found nontrivial examples where Kronecker conjugacy holds. Until now there were only finitely many known such examples. This paper provides the first infinite series. The main part of the construction is group theoretic.
 1.
Michael
D. Fried and Moshe
Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer
Grenzgebiete (3) [Results in Mathematics and Related Areas (3)],
vol. 11, SpringerVerlag, Berlin, 1986. MR 868860
(89b:12010)
 2.
Robert
M. Guralnick, Zeroes of permutation characters with applications to
prime splitting and Brauer groups, J. Algebra 131
(1990), no. 1, 294–302. MR 1055010
(91j:20038), http://dx.doi.org/10.1016/00218693(90)90177P
 3.
Wolfram
Jehne, Kronecker classes of algebraic number fields, J. Number
Theory 9 (1977), no. 3, 279–320. MR 0447184
(56 #5499)
 4.
L. KRONECKER. Über die Irreduzibilität von Gleichungen. Werke II, 8593; Monatsberichte Deutsche Akademie für Wissenschaft (1880), 155163.
 5.
G. MALLE, B. H. MATZAT. Inverse Galois Theory. Book manuscript.
 6.
P. MÜLLER. Primitive monodromy groups of polynomials. Contemp. Math. 186 (1995), 385401. CMP 96:01
 7.
P. MÜLLER. Kronecker conjugacy of polynomials. Preprint (1995).
 8.
Cheryl
E. Praeger, Kronecker classes of field extensions of small
degree, J. Austral. Math. Soc. Ser. A 50 (1991),
no. 2, 297–315. MR 1094925
(92m:12004)
 9.
J. F. RITT. Prime and composite polynomials. Trans. Amer. Math. Soc. 23 (1922), 5166.
 10.
Jan
Saxl, On a question of W. Jehne concerning covering subgroups of
groups and Kronecker classes of fields, J. London Math. Soc. (2)
38 (1988), no. 2, 243–249. MR 966296
(90b:11118), http://dx.doi.org/10.1112/jlms/s238.2.243
 11.
H. VÖLKLEIN. Groups as Galois Groups  an Introduction. Cambridge University Press, 1996.
 1.
 M. FRIED, M. JARDEN. Field Arithmetic. Springer, Berlin Heidelberg, 1986. MR 89b:12010
 2.
 R. GURALNICK. Zeroes of permutation characters with applications to prime splitting and Brauer groups. J. Algebra 131 (1990), 294302. MR 91j:20038
 3.
 W. JEHNE. Kronecker classes of algebraic number fields. J. Number Theory 9 (1977), 279320. MR 56:5499
 4.
 L. KRONECKER. Über die Irreduzibilität von Gleichungen. Werke II, 8593; Monatsberichte Deutsche Akademie für Wissenschaft (1880), 155163.
 5.
 G. MALLE, B. H. MATZAT. Inverse Galois Theory. Book manuscript.
 6.
 P. MÜLLER. Primitive monodromy groups of polynomials. Contemp. Math. 186 (1995), 385401. CMP 96:01
 7.
 P. MÜLLER. Kronecker conjugacy of polynomials. Preprint (1995).
 8.
 C. PRAEGER. Kronecker classes of field extensions of small degree. J. Austr. Math. Soc. (Series A) 50 (1991), 297315. MR 92m:12004
 9.
 J. F. RITT. Prime and composite polynomials. Trans. Amer. Math. Soc. 23 (1922), 5166.
 10.
 J. SAXL. On a question of W. Jehne concerning covering subgroups of groups and Kronecker classes of fields. J. London. Math. Soc.(2) 38 (1988), 243249. MR 90b:11118
 11.
 H. VÖLKLEIN. Groups as Galois Groups  an Introduction. Cambridge University Press, 1996.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
11C08,
11R09,
20B05,
11R32,
12E05,
12F10
Retrieve articles in all journals
with MSC (1991):
11C08,
11R09,
20B05,
11R32,
12E05,
12F10
Additional Information
Peter Müller
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email:
pfm@math.ufl.edu
DOI:
http://dx.doi.org/10.1090/S0002993997038926
PII:
S 00029939(97)038926
Received by editor(s):
January 18, 1996
Additional Notes:
The author thanks the Deutsche Forschungsgemeinschaft (DFG) for its support in form of a postdoctoral fellowship.
Communicated by:
William W. Adams
Article copyright:
© Copyright 1997
American Mathematical Society
