An infinite series

of Kronecker conjugate polynomials

Author:
Peter Müller

Journal:
Proc. Amer. Math. Soc. **125** (1997), 1933-1940

MSC (1991):
Primary 11C08, 11R09, 20B05; Secondary 11R32, 12E05, 12F10

MathSciNet review:
1396989

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Abstract: Let be a field of characteristic , a transcendental over , and be the absolute Galois group of . Then two non-constant 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 non-zero 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 non-trivial 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, Springer-Verlag, Berlin, 1986. MR**868860****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**, 10.1016/0021-8693(90)90177-P**3.**Wolfram Jehne,*Kronecker classes of algebraic number fields*, J. Number Theory**9**(1977), no. 3, 279–320. MR**0447184****4.**L. KRONECKER.*Über die Irreduzibilität von Gleichungen.*Werke II, 85-93; Monatsberichte Deutsche Akademie für Wissenschaft (1880), 155-163.**5.**G. MALLE, B. H. MATZAT.*Inverse Galois Theory.*Book manuscript.**6.**P. MÜLLER.*Primitive monodromy groups of polynomials.*Contemp. Math. 186 (1995), 385-401. 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****9.**J. F. RITT.*Prime and composite polynomials.*Trans. Amer. Math. Soc. 23 (1922), 51-66.**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**, 10.1112/jlms/s2-38.2.243**11.**H. VÖLKLEIN.*Groups as Galois Groups - an Introduction.*Cambridge University Press, 1996.

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Additional Information

**Peter Müller**

Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611

Email:
pfm@math.ufl.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-03892-6

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