Abstract
Let $k$ be a field of characteristic $p>0$, let $q$ be a power of $p$, and let $u$ be transcendental over $k$. We determine all polynomials $f\in k[X]\setminus k[X^p]$ of degree $q(q-1)/2$ for which the Galois group of $f(X)-u$ over $k(u)$ has a transitive normal subgroup isomorphic to $\operatorname{PSL}_2(q)$, subject to a certain ramification hypothesis. As a consequence, we describe all polynomials $f\in k[X]$ such that $\operatorname{deg}(f)$ is not a power of $p$ and $f$ is functionally indecomposable over $k$ but $f$ decomposes over an extension of $k$. Moreover, except for one ramification configuration (which is handled in a companion paper with Rosenberg), we describe all indecomposable polynomials $f\in k[X]$ such that $\operatorname{deg}(f)$ is not a power of $p$ and $f$ is exceptional, in the sense that $X-Y$ is the only absolutely irreducible factor of $f(X)-f(Y)$ which lies in $k[X,Y]$. It is known that, when $k$ is finite, a polynomial $f$ is exceptional if and only if it induces a bijection on infinitely many finite extensions of $k$.
-
[BZ] R. M. Beals and M. E. Zieve, Decompositions of polynomials.
@misc{BZ,
author={Beals, R. M. and Zieve, M. E.},
TITLE={Decompositions of polynomials},
NOTE={preprint},
} -
[Ca] R. W. Carter, Simple Groups of Lie Type, New York: John Wiley & Sons, 1972, vol. 28.
@book {Ca, MRKEY = {0407163},
AUTHOR = {Carter, Roger W.},
TITLE = {Simple Groups of {L}ie Type},
SERIES= {Pure Appl. Math.},
VOLUME={28},
PUBLISHER = {John Wiley \& Sons},
ADDRESS={New York},
YEAR = {1972},
PAGES = {viii+331},
MRCLASS = {20G15 (20D05 22E20)},
MRNUMBER = {53 \#10946},
MRREVIEWER = {Louis Solomon},
ZBLNUMBER = {0248.20015},
} -
[Coh] S. D. Cohen, "The distribution of polynomials over finite fields," Acta Arith., vol. 17, pp. 255-271, 1970.
@article {Coh, MRKEY = {0277501},
AUTHOR = {Cohen, Stephen D.},
TITLE = {The distribution of polynomials over finite fields},
JOURNAL = {Acta Arith.},
FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
VOLUME = {17},
YEAR = {1970},
PAGES = {255--271},
ISSN = {0065-1036},
MRCLASS = {10.77 (12.00)},
MRNUMBER = {43 \#3234},
MRREVIEWER = {H. W. Brinkmann},
ZBLNUMBER = {0209.36001},
} -
[Co2p]
S. D. Cohen, "Permutation polynomials and primitive permutation groups," Arch. Math. $($Basel$)$, vol. 57, iss. 5, pp. 417-423, 1991.
@article {Co2p, MRKEY = {1129514},
AUTHOR = {Cohen, Stephen D.},
TITLE = {Permutation polynomials and primitive permutation groups},
JOURNAL = {Arch. Math. $($Basel$)$},
FJOURNAL = {Archiv der Mathematik},
VOLUME = {57},
YEAR = {1991},
NUMBER = {5},
PAGES = {417--423},
ISSN = {0003-889X},
CODEN = {ACVMAL},
MRCLASS = {11T06 (20B15)},
MRNUMBER = {92j:11145},
MRREVIEWER = {Gary L. Mullen},
DOI = {10.1007/BF01246737},
ZBLNUMBER = {0766.11047},
} -
[CM]
S. D. Cohen and R. W. Matthews, "A class of exceptional polynomials," Trans. Amer. Math. Soc., vol. 345, iss. 2, pp. 897-909, 1994.
@article {CM, MRKEY = {1272675},
AUTHOR = {Cohen, Stephen D. and Matthews, Rex W.},
TITLE = {A class of exceptional polynomials},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {345},
YEAR = {1994},
NUMBER = {2},
PAGES = {897--909},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {11T06},
MRNUMBER = {95d:11164},
MRREVIEWER = {Gary L. Mullen},
DOI = {10.2307/2155005},
ZBLNUMBER = {0812.11070},
} -
[Dith]
L. E. Dickson, "The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group," Ann. of Math., vol. 11, iss. 1-6, pp. 65-120, 1896/97.
@article {Dith, MRKEY = {1502214},
AUTHOR = {Dickson, Leonard Eugene},
TITLE = {The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics},
VOLUME = {11},
YEAR = {1896/97},
NUMBER = {1-6},
PAGES = {65--120},
ISSN = {0003-486X},
MRCLASS = {Contributed Item},
MRNUMBER = {1502214},
DOI = {10.2307/1967217},
JFMNUMBER={28.0135.03},
} -
[Di] L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory, Leipzig: Teubner, 1901.
@book{Di,
author={Dickson, Leonard Eugene},
TITLE={Linear Groups with an Exposition of the Galois Field Theory},
PUBLISHER={Teubner},
ADDRESS={Leipzig},
YEAR={1901},
JFMNUMBER={32.0128.01},
} -
[Di2]
L. E. Dickson, "An invariantive investigation of irreducible binary modular forms," Trans. Amer. Math. Soc., vol. 12, iss. 1, pp. 1-18, 1911.
@article {Di2, MRKEY = {1500877},
AUTHOR = {Dickson, Leonard Eugene},
TITLE = {An invariantive investigation of irreducible binary modular forms},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {12},
YEAR = {1911},
NUMBER = {1},
PAGES = {1--18},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {13A50},
MRNUMBER = {1500877},
DOI = {10.2307/1988731},
JFMNUMBER={42.0134.02},
} -
[DM] J. D. Dixon and B. Mortimer, Permutation Groups, New York: Springer-Verlag, 1996, vol. 163.
@book {DM, MRKEY = {1409812},
AUTHOR = {Dixon, John D. and Mortimer, Brian},
TITLE = {Permutation Groups},
SERIES = {Grad. Texts Math.},
VOLUME = {163},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1996},
PAGES = {xii+346},
ISBN = {0-387-94599-7},
MRCLASS = {20B05 (20-01 20B07)},
MRNUMBER = {98m:20003},
MRREVIEWER = {Martin W. Liebeck},
ZBLNUMBER = {0951.20001},
} -
[E] N. D. Elkies, "Linearized algebra and finite groups of Lie type. I. Linear and symplectic groups," in Applications of Curves over Finite Fields, Providence, RI: Amer. Math. Soc., 1999, vol. 245, pp. 77-107.
@incollection {E, MRKEY = {1732230},
AUTHOR = {Elkies, Noam D.},
TITLE = {Linearized algebra and finite groups of {L}ie type. {I}. {L}inear and symplectic groups},
BOOKTITLE = {Applications of Curves over Finite Fields},
VENUE={{S}eattle, {WA},
1997},
SERIES = {Contemp. Math.},
VOLUME= {245},
PAGES = {77--107},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1999},
MRCLASS = {20G40 (11G25 12F12 13A50)},
MRNUMBER = {2001a:20082},
MRREVIEWER = {R. W. Carter},
ZBLNUMBER={0976.20032},
} -
[Fe] W. Feit, "On symmetric balanced incomplete block designs with doubly transitive automorphism groups," J. Combinatorial Theory Ser. A, vol. 14, pp. 221-247, 1973.
@article {Fe, MRKEY = {0327540},
AUTHOR = {Feit, Walter},
TITLE = {On symmetric balanced incomplete block designs with doubly transitive automorphism groups},
JOURNAL = {J. Combinatorial Theory Ser. A},
FJOURNAL = {Journal of Combinatorial Theory. Series A},
VOLUME = {14},
YEAR = {1973},
PAGES = {221--247},
MRCLASS = {05B05},
MRNUMBER = {48 \#5882},
MRREVIEWER = {A. Gewirtz},
ZBLNUMBER = {0278.05016},
} -
[Fe2] W. Feit, "Some consequences of the classification of finite simple groups," in The Santa Cruz Conference on Finite Groups, Providence, R.I.: Amer. Math. Soc., 1980, vol. 37, pp. 175-181.
@incollection {Fe2, MRKEY = {604576},
AUTHOR = {Feit, Walter},
TITLE = {Some consequences of the classification of finite simple groups},
BOOKTITLE = {The {S}anta {C}ruz {C}onference on {F}inite {G}roups},
VENUE={{U}niv. {C}alifornia, {S}anta {C}ruz, {C}alif., 1979},
SERIES = {Proc. Sympos. Pure Math.},
VOLUME= {37},
PAGES = {175--181},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, R.I.},
YEAR = {1980},
MRCLASS = {20C20 (20D05)},
MRNUMBER = {82c:20019},
MRREVIEWER = {Henry S. Leonard, Jr.},
ZBLNUMBER = {0454.20014},
} -
@article {Fr, MRKEY = {0257033},
AUTHOR = {Fried, Michael},
TITLE = {On a conjecture of {S}chur},
JOURNAL = {Michigan Math. J.},
FJOURNAL = {The Michigan Mathematical Journal},
VOLUME = {17},
YEAR = {1970},
PAGES = {41--55},
ISSN = {0026-2285},
MRCLASS = {10.76},
MRNUMBER = {41 \#1688},
MRREVIEWER = {A. L. Whiteman},
URL = {http://projecteuclid.org/getRecord?id=euclid.mmj/1029000374},
ZBLNUMBER = {0169.37702},
} -
[FGS]
M. D. Fried, R. Guralnick, and J. Saxl, "Schur covers and Carlitz’s conjecture," Israel J. Math., vol. 82, iss. 1-3, pp. 157-225, 1993.
@article {FGS, MRKEY = {1239049},
AUTHOR = {Fried, Michael D. and Guralnick, Robert and Saxl, Jan},
TITLE = {Schur covers and {C}arlitz's conjecture},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {82},
YEAR = {1993},
NUMBER = {1-3},
PAGES = {157--225},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {12E20 (11T06 20D06)},
MRNUMBER = {94j:12007},
MRREVIEWER = {Martin W. Liebeck},
DOI = {10.1007/BF02808112},
ZBLNUMBER = {0855.11063},
} -
[GaSt]
A. Garc’ia and H. Stichtenoth, "Elementary abelian $p$-extensions of algebraic function fields," Manuscripta Math., vol. 72, iss. 1, pp. 67-79, 1991.
@article {GaSt, MRKEY = {1107453},
AUTHOR = {Garc{\'ı}a, Arnaldo and Stichtenoth, Henning},
TITLE = {Elementary abelian {$p$}-extensions of algebraic function fields},
JOURNAL = {Manuscripta Math.},
FJOURNAL = {Manuscripta Mathematica},
VOLUME = {72},
YEAR = {1991},
NUMBER = {1},
PAGES = {67--79},
ISSN = {0025-2611},
CODEN = {MSMHB2},
MRCLASS = {11R58 (14H05)},
MRNUMBER = {92j:11139},
MRREVIEWER = {Paulo Viana},
DOI = {10.1007/BF02568266},
ZBLNUMBER = {0739.14015},
} -
[GL] D. Gorenstein and R. Lyons, "The local structure of finite groups of characteristic $2$ type," Mem. Amer. Math. Soc., vol. 42, iss. 276, p. vii, 1983.
@article {GL, MRKEY = {690900},
AUTHOR = {Gorenstein, Daniel and Lyons, Richard},
TITLE = {The local structure of finite groups of characteristic {$2$} type},
JOURNAL = {Mem. Amer. Math. Soc.},
FJOURNAL = {Memoirs of the American Mathematical Society},
VOLUME = {42},
YEAR = {1983},
NUMBER = {276},
PAGES = {vii+731},
ISSN = {0065-9266},
CODEN = {MAMCAU},
MRCLASS = {20D05},
MRNUMBER = {84g:20025},
MRREVIEWER = {Stephen D. Smith},
ZBLNUMBER={0519.20014},
} -
[GM]
R. M. Guralnick and P. Müller, "Exceptional polynomials of affine type," J. Algebra, vol. 194, iss. 2, pp. 429-454, 1997.
@article {GM, MRKEY = {1467161},
AUTHOR = {Guralnick, Robert M. and M{ü}ller, Peter},
TITLE = {Exceptional polynomials of affine type},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {194},
YEAR = {1997},
NUMBER = {2},
PAGES = {429--454},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {12F20 (11T06 20G40)},
MRNUMBER = {98i:12006},
MRREVIEWER = {S. D. Cohen},
DOI = {10.1006/jabr.1997.7028},
ZBLNUMBER={0911.11049},
} -
[GMZ] R. M. Guralnick, P. Müller, and M. Zieve, Exceptional polynomials of affine type, revisited.
@misc{GMZ,
author={Guralnick, Robert M. and M{ü}ller, Peter and Zieve, M.},
TITLE={Exceptional polynomials of affine type, revisited},
NOTE={preprint},
} -
[GRZ] R. M. Guralnick, J. E. Rosenberg, and M. E. Zieve, "A new family of exceptional polynomials in characteristic two," Ann. of Math., vol. 172, pp. 1361-1390, 2010.
@article{GRZ,
author={Guralnick, Robert M. and Rosenberg, J. E. and Zieve, M. E.},
TITLE={A new family of exceptional polynomials in characteristic two},
JOURNAL={Ann. of Math.},
VOLUME={172},
YEAR={2010},
PAGES={1361--1390},
} -
[GRZ2] R. M. Guralnick, J. E. Rosenberg, and M. E. Zieve, A classification of polynomials with $\operatorname{PSL}(2)$ monodromy.
@misc{GRZ2,
author={Guralnick, Robert M. and Rosenberg, J. E. and Zieve, M. E.},
TITLE={A classification of polynomials with $\operatorname{PSL}(2)$ monodromy},
NOTE={preprint},
} -
[GS]
R. M. Guralnick and J. Saxl, "Monodromy groups of polynomials," in Groups of Lie type and their Geometries, Cambridge: Cambridge Univ. Press, 1995, vol. 207, pp. 125-150.
@incollection {GS, MRKEY = {1320519},
AUTHOR = {Guralnick, Robert M. and Saxl, Jan},
TITLE = {Monodromy groups of polynomials},
BOOKTITLE = {Groups of {L}ie type and their Geometries},
VENUE={{C}omo, 1993},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME = {207},
PAGES = {125--150},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1995},
MRCLASS = {20B15 (12F10 14H05)},
MRNUMBER = {96b:20003},
MRREVIEWER = {Ulrich Dempwolff},
DOI = {10.1017/CBO9780511565823.012},
ZBLNUMBER = {0848.12002},
} -
[GS2] R. M. Guralnick and J. Saxl, "Exceptional polynomials over arbitrary fields," in Algebra, Arithmetic and Geometry with Applications, New York: Springer-Verlag, 2004, pp. 457-472.
@incollection {GS2, MRKEY = {2037103},
AUTHOR = {Guralnick, Robert M. and Saxl, Jan},
TITLE = {Exceptional polynomials over arbitrary fields},
BOOKTITLE = {Algebra, Arithmetic and Geometry with Applications},
VENUE={{W}est {L}afayette, {IN},
2000},
PAGES = {457--472},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2004},
MRCLASS = {12F10 (14H30)},
MRNUMBER = {2005h:12007},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {1117.12300},
} -
[GTZ] R. M. Guralnick, T. J. Tucker, and M. E. Zieve, "Exceptional covers and bijections on rational points," Int. Math. Res. Not., vol. 2007 , no. 1, p. I.
@article {GTZ, MRKEY = {2331902},
AUTHOR = {Guralnick, Robert M. and Tucker, Thomas J. and Zieve, Michael E.},
TITLE = {Exceptional covers and bijections on rational points},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices. IMRN},
PAGES = {Art. ID rnm004, 20},
ISSN = {1073-7928},
MRCLASS = {14G15},
MRNUMBER = {2008e:14029},
MRREVIEWER = {Liang-Chung Hsia},
VOLUME = {2007{\rm , no. 1}},
ZBLNUMBER={1127.14023},
} -
[GZ2] R. M. Guralnick and M. E. Zieve, Automorphisms of ordinary curves.
@misc{GZ2,
author = {Guralnick, Robert M. and Zieve, Michael E.},
TITLE={Automorphisms of ordinary curves},
NOTE={in preparation},
} -
[har]
D. Harbater, "Abhyankar’s conjecture on Galois groups over curves," Invent. Math., vol. 117, iss. 1, pp. 1-25, 1994.
@article {har, MRKEY = {1269423},
AUTHOR = {Harbater, David},
TITLE = {Abhyankar's conjecture on {G}alois groups over curves},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {117},
YEAR = {1994},
NUMBER = {1},
PAGES = {1--25},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {14H30 (14E20)},
MRNUMBER = {95i:14029},
MRREVIEWER = {Wolfgang K. Seiler},
DOI = {10.1007/BF01232232},
ZBLNUMBER = {0805.14014},
} -
[Kl] A. A. Klyachko, "Monodromy groups of polynomial mappings," Stud. in Number Theory, vol. 6, pp. 82-91, 1975.
@article{Kl,
author={Klyachko, A. A.},
TITLE={Monodromy groups of polynomial mappings},
JOURNAL={Stud. in Number Theory},
VOLUME={6},
YEAR={1975},
PAGES={82--91},
NOTE={in Russian},
ZBLNUMBER={0408.12023},
} -
[LMZ] H. W. Lenstra Jr., D. P. Moulton, and M. Zieve, Exceptional maps between varieties.
@misc{LMZ,
author = {Lenstra, Jr., H. W. and Moulton, D. P. and Zieve, M.},
TITLE={Exceptional maps between varieties},
NOTE={in preparation},
} -
[LZ]
H. W. Lenstra Jr. and M. Zieve, "A family of exceptional polynomials in characteristic three," in Finite Fields and Applications, Cambridge: Cambridge Univ. Press, 1996, vol. 233, pp. 209-218.
@incollection {LZ, MRKEY = {1433150},
AUTHOR = {Lenstra, Jr., H. W. and Zieve, M.},
TITLE = {A family of exceptional polynomials in characteristic three},
BOOKTITLE = {Finite Fields and Applications },
VENUE={{G}lasgow, 1995},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME= {233},
PAGES = {209--218},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1996},
MRCLASS = {11T06 (20G40)},
MRNUMBER = {98a:11174},
MRREVIEWER = {S. D. Cohen},
DOI = {10.1017/CBO9780511525988.018},
ZBLNUMBER = {0871.11087},
} -
[Le]
H. Leopoldt, "Über die Automorphismengruppe des Fermatkörpers," J. Number Theory, vol. 56, iss. 2, pp. 256-282, 1996.
@article {Le, MRKEY = {1373551},
AUTHOR = {Leopoldt, Heinrich-Wolfgang},
TITLE = {Über die {A}utomorphismengruppe des {F}ermatkörpers},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {56},
YEAR = {1996},
NUMBER = {2},
PAGES = {256--282},
ISSN = {0022-314X},
CODEN = {JNUTA9},
MRCLASS = {11G20 (14H25 14H45)},
MRNUMBER = {96k:11079},
MRREVIEWER = {Ernst-Ulrich Gekeler},
DOI = {10.1006/jnth.1996.0017},
ZBLNUMBER={0854.11062},
} -
[Lu] H. Lüneburg, Translation Planes, New York: Springer-Verlag, 1980.
@book {Lu, MRKEY = {572791},
AUTHOR = {L{ü}neburg, Heinz},
TITLE = {Translation Planes},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1980},
PAGES = {ix+278},
ISBN = {3-540-09614-0},
MRCLASS = {51A40 (51-02)},
MRNUMBER = {83h:51008},
MRREVIEWER = {J. Andr{é}},
ZBLNUMBER={0446.51003},
} -
[Mac] C. R. MacCluer, "On a conjecture of Davenport and Lewis concerning exceptional polynomials," Acta Arith., vol. 12, pp. 289-299, 1966/1967.
@article {Mac, MRKEY = {0207638},
AUTHOR = {MacCluer, C. R.},
TITLE = {On a conjecture of {D}avenport and {L}ewis concerning exceptional polynomials},
JOURNAL = {Acta Arith.},
FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
VOLUME = {12},
YEAR = {1966/1967},
PAGES = {289--299},
ISSN = {0065-1036},
MRCLASS = {12.30 (10.67)},
MRNUMBER = {34 \#7453},
MRREVIEWER = {A. L. Whiteman},
ZBLNUMBER = {0171.00501},
} -
[Mu] P. Müller, "New examples of exceptional polynomials," in Finite Fields: Theory, Applications, and Algorithms, Providence, RI: Amer. Math. Soc., 1994, vol. 168, pp. 245-249.
@incollection {Mu, MRKEY = {1291433},
AUTHOR = {M{ü}ller, Peter},
TITLE = {New examples of exceptional polynomials},
BOOKTITLE = {Finite Fields: Theory, Applications, and Algorithms},
VENUE={{L}as {V}egas, {NV},
1993},
SERIES = {Contemp. Math.},
VOLUME = {168},
PAGES = {245--249},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1994},
MRCLASS = {11T06 (11R09 20B15)},
MRNUMBER = {95h:11136},
MRREVIEWER = {S. D. Cohen},
ZBLNUMBER={0818.11047},
} -
[Mumon] P. Müller, "Primitive monodromy groups of polynomials," in Recent Developments in the Inverse Galois Problem, Providence, RI: Amer. Math. Soc., 1995, vol. 186, pp. 385-401.
@incollection {Mumon, MRKEY = {1352284},
AUTHOR = {M{ü}ller, Peter},
TITLE = {Primitive monodromy groups of polynomials},
BOOKTITLE = {Recent Developments in the Inverse {G}alois Problem},
VENUE={{S}eattle, {WA},
1993},
SERIES = {Contemp. Math.},
VOLUME = {186},
PAGES = {385--401},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1995},
MRCLASS = {20B15 (11C08 12F10 20B20)},
MRNUMBER = {96m:20004},
MRREVIEWER = {Robert M. Guralnick},
ZBLNUMBER={0840.12001},
} -
[MuSchur]
P. Müller, "A Weil-bound free proof of Schur’s conjecture," Finite Fields Appl., vol. 3, iss. 1, pp. 25-32, 1997.
@article {MuSchur, MRKEY = {1429041},
AUTHOR = {M{ü}ller, Peter},
TITLE = {A {W}eil-bound free proof of {S}chur's conjecture},
JOURNAL = {Finite Fields Appl.},
FJOURNAL = {Finite Fields and their Applications},
VOLUME = {3},
YEAR = {1997},
NUMBER = {1},
PAGES = {25--32},
ISSN = {1071-5797},
MRCLASS = {11R58 (11T06)},
MRNUMBER = {98d:11143},
MRREVIEWER = {G. Turnwald},
DOI = {10.1006/ffta.1996.0170},
ZBLNUMBER={0904.11040},
} -
[ray]
M. Raynaud, "Revêtements de la droite affine en caractéristique $p>0$ et conjecture d’Abhyankar," Invent. Math., vol. 116, iss. 1-3, pp. 425-462, 1994.
@article {ray, MRKEY = {1253200},
AUTHOR = {Raynaud, M.},
TITLE = {Revêtements de la droite affine en caractéristique {$p>0$} et conjecture d'{A}bhyankar},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {116},
YEAR = {1994},
NUMBER = {1-3},
PAGES = {425--462},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {14H30 (14E20)},
MRNUMBER = {94m:14034},
MRREVIEWER = {Wolfgang K. Seiler},
DOI = {10.1007/BF01231568},
ZBLNUMBER = {0798.14013},
} -
[Se] J. Serre, Local fields, New York: Springer-Verlag, 1979, vol. 67.
@book {Se, MRKEY = {554237},
AUTHOR = {Serre, Jean-Pierre},
TITLE = {Local fields},
SERIES = {Grad. Texts Math.},
VOLUME= {67},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1979},
PAGES = {viii+241},
ISBN = {0-387-90424-7},
MRCLASS = {12Bxx},
MRNUMBER = {82e:12016},
ZBLNUMBER = {0423.12016},
} -
[Si] J. H. Silverman, The Arithmetic of Elliptic Curves, New York: Springer-Verlag, 1986, vol. 106.
@book {Si, MRKEY = {817210},
AUTHOR = {Silverman, Joseph H.},
TITLE = {The Arithmetic of Elliptic Curves},
SERIES = {Grad. Texts Math.},
VOLUME = {106},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1986},
PAGES = {xii+400},
ISBN = {0-387-96203-4},
MRCLASS = {11G05 (14Gxx 14K07 14K15)},
MRNUMBER = {87g:11070},
MRREVIEWER = {Robert S. Rumely},
ZBLNUMBER = {0585.14026},
} -
[St2]
H. Stichtenoth, "Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. II. Ein spezieller Typ von Funktionenkörpern," Arch. Math. $($Basel$)$, vol. 24, pp. 615-631, 1973.
@article {St2, MRKEY = {0404265},
AUTHOR = {Stichtenoth, Henning},
TITLE = {Über die {A}utomorphismengruppe eines algebraischen {F}unktionenkörpers von {P}rimzahlcharakteristik. {II}. {E}in spezieller {T}yp von {F}unktionenkörpern},
JOURNAL = {Arch. Math. $($Basel$)$},
FJOURNAL = {Archiv der Mathematik},
VOLUME = {24},
YEAR = {1973},
PAGES = {615--631},
ISSN = {0003-889X},
MRCLASS = {14H05},
MRNUMBER = {53 \#8068},
MRREVIEWER = {W.-D. Geyer},
DOI = {10.1007/BF01228261},
} -
[St] H. Stichtenoth, Algebraic Function Fields and Codes, New York: Springer-Verlag, 1993.
@book {St, MRKEY = {1251961},
AUTHOR = {Stichtenoth, Henning},
TITLE = {Algebraic Function Fields and Codes},
SERIES = {Universitext},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1993},
PAGES = {x+260},
ISBN = {3-540-56489-6},
MRCLASS = {14H05 (11R58 11T71 14G15 94B27)},
MRNUMBER = {94k:14016},
MRREVIEWER = {R. F. Lax},
ZBLNUMBER = {0816.14011},
} -
[Su] M. Suzuki, Group Theory. I, New York: Springer-Verlag, 1982, vol. 247.
@book {Su, MRKEY = {648772},
AUTHOR = {Suzuki, Michio},
TITLE = {Group Theory. {\rm I}},
SERIES = {Grundl. Math. Wissen.},
VOLUME = {247},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1982},
PAGES = {xiv+434},
ISBN = {3-540-10915-3},
MRCLASS = {20-01},
MRNUMBER = {82k:20001c},
MRREVIEWER = {B. Chang},
ZBLNUMBER = {0472.20001},
} -
[W2p]
D. Q. Wan, "Permutation polynomials and resolution of singularities over finite fields," Proc. Amer. Math. Soc., vol. 110, iss. 2, pp. 303-309, 1990.
@article {W2p, MRKEY = {1031673},
AUTHOR = {Wan, Da Qing},
TITLE = {Permutation polynomials and resolution of singularities over finite fields},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {110},
YEAR = {1990},
NUMBER = {2},
PAGES = {303--309},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {11T06},
MRNUMBER = {90m:11197},
MRREVIEWER = {Gary L. Mullen},
DOI = {10.2307/2048070},
ZBLNUMBER = {0711.11050},
} -
[Z]
M. Zieve, "Bivariate factorizations via Galois theory, with application to exceptional polynomials," J. Algebra, vol. 210, iss. 2, pp. 670-689, 1998.
@article {Z, MRKEY = {1662312},
AUTHOR = {Zieve, Michael},
TITLE = {Bivariate factorizations via {G}alois theory, with application to exceptional polynomials},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {210},
YEAR = {1998},
NUMBER = {2},
PAGES = {670--689},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {12E05 (12F10 13P05)},
MRNUMBER = {99m:12001},
MRREVIEWER = {Fernando Barrera-Mora},
DOI = {10.1006/jabr.1998.7581},
ZBLNUMBER = {0922.12010},
} -
[Zs]
K. Zsigmondy, "Zur Theorie der Potenzreste," Monatsh. Math. Phys., vol. 3, iss. 1, pp. 265-284, 1892.
@article {Zs, MRKEY = {1546236},
AUTHOR = {Zsigmondy, K.},
TITLE = {Zur {T}heorie der {P}otenzreste},
JOURNAL = {Monatsh. Math. Phys.},
FJOURNAL = {Monatshefte für Mathematik und Physik},
VOLUME = {3},
YEAR = {1892},
NUMBER = {1},
PAGES = {265--284},
ISSN = {0026-9255},
MRCLASS = {Contributed Item},
MRNUMBER = {1546236},
DOI = {10.1007/BF01692444},
}