Regulators of an Infinite Family of theSimplest Quartic Function Fields

Lee, Jungyun ; Lee, Yoonjin

Canadian journal of mathematics, Tome 69 (2017) no. 3, pp. 579-594

Résumé

We explicitly find regulators of an infinite family $\{{{L}_{m}}\}$ of the simplest quartic function fields with a parameter $m$ in a polynomial ring ${{\mathbb{F}}_{q}}\left( t \right)$ , where ${{\mathbb{F}}_{q}}$ is the finite field of order $q$ with odd characteristic. In fact, this infinite family of the simplest quartic function fields are subfields of maximal real subfields of cyclotomic function fields having the same conductors. We obtain a lower bound on the class numbers of the family $\{{{L}_{m}}\}$ and some result on the divisibility of the divisor class numbers of cyclotomic function fields that contain $\{{{L}_{m}}\}$ as their subfields. Furthermore, we find an explicit criterion for the characterization of splitting types of all the primes of the rational function field ${{\mathbb{F}}_{q}}\left( t \right)$ in $\{{{L}_{m}}\}$ .

Détail

DOI : 10.4153/CJM-2016-038-2

Mots-clés : 11R29, 11R58, regulator, function field, quartic extension, class number

Bibliographie

[1] [1] Bae, S., Real quadratic function fields of Richaud-Degert type with ideal class number one. Proc. Amer. Soc. Math. 140(2011), no. 2, 403–414. Google Scholar | DOI

[2] [2] Cassels, J. W. S., An introduction to the geometry of numbers. Springer-Verlag, Berlin, 1971. Google Scholar

[3] [3] Feng, K. and Hu, W., On real quadratic function fields of Chowla type with ideal class number one. Proc. Amer. Soc. Math. 127(1999), no. 5,1301–1307. Google Scholar | DOI

[4] [4] Gras, M-N., Table numérique du nombre de classes et des unités des extensions cycliques réelles de degré 4 de ℚ. Publ. Math. Fac. Sci. Besançon, Théorie des Nombres, Fasc. 2, (1977/78) 1–53. Google Scholar

[5] [5] Gras, M-N., Special units in real cyclic sextic fields. Math. Comp. 48(1987), no. 177,179–182. Google Scholar | DOI

[6] [6] Guo, L. and Shu, L., Class numbers of cyclotomic function fields. Trans. Amer. Math. Soc. 351(1999), no. 11, 4445–4467. Google Scholar | DOI

[7] [7] Haghighi, M., Computation of conductor for finite extensions with cyclic Galois groups. J. Algebra 124(1989), no. 2, 329–333. Google Scholar | DOI

[8] [8] Hasse, H., Arithmetische Bestimmung von Grundeinheit und Klassenzahl in zyklischen kubischen und biquadratischen Zahlkörpern, Mathematische Abhandlungen. Band 3, Walter de Gruyter, Berlin, 1975, pp. 290–379. Google Scholar

[9] [9] Im, B-H. and Lee, Y., Decomposition of places in dihedral and cyclic quintic trinomial extensions of global fields. Manuscripta Math. 137(2012), no. 1-2,107–127. Google Scholar | DOI

[10] [10] Kishi, Y., A family of cyclic cubic polynomials whose roots are systems of fundamental units. J. Number Theory 102(2003), no. 1, 90–106. http://dx.doi.Org/10.1016/S0022-314X(03)00085-4 Google Scholar

[11] [11] Lazarus, A. J., On the class number and unit index of simplest quartic fields. Nagoya Math. J. 121(1991), 1–13. Google Scholar | DOI

[12] [12] Lee, Y., Scheidler, R., and Yarrish, C., Computation of the fundamental units and the regulator of a cyclic cubic function field. Experiment. Math. 12(2003), no. 2, 211–225. Google Scholar | DOI

[13] [13] Lehmer, E., Connection between Gaussian periods and cyclic units. Math. Comp. 50(1988), no. 182, 535–541. Google Scholar | DOI

[14] [14] Louboutin, S. R., Hasse unit indices of dihedral octic CM-fields. Math. Nachr. 215(2000), 107–113. http://dx.doi.Org/10.1002/1522-2616(200007)215:1<107::AID-MANA107>3.0.CO;2-</107::AID-MANA107> 3.0.CO;2->Google Scholar

[15] [15] Louboutin, S. R., The simplest quartic fields with ideal class groups of exponents less than or equal to 2. J. Math. Soc. Japan 56(2004), no. 3, 717–727. http://dx.doi.Org/10.2969/jmsj71191334082 Google Scholar

[16] [16] Rosen, M., Number theory in function fields. Graduate Texts in Mathematics, 210. Springer-Verlag, New York, 2002. http://dx.doi.Org/10.1007/978-1-4757-6046-0, Google Scholar

[17] [17] Scheidler, R. and Stein, A., Approximating Euler products and class number computation in algebraic function fields. Rocky Mountain J. Math. 40(2010), no. 5,1689–1727. http://dx.doi.Org/10.1216/RMJ-2O10-40-5-1689 Google Scholar

[18] [18] Schoof, R. and Washington, L. C., Quintic polynomials and real cyclotomic fields with large class numbers. Math. Comp. 50(1988), no. 182, 543–556. http://dx.doi.Org/10.2307/2008623 Google Scholar

[19] [19] Shen, Y. Y, Unit groups and class numbers of real cyclic octic fields. Trans. Amer. Math. Soc. 326(1991), no. 1, 179–209. Google Scholar | DOI

[20] [20] Stein, A. and Williams, H. C., Some methods for evaluating the regulator of a real quadratic function field. Experiment. Math. 8(1999), no. 2,119–133. Google Scholar | DOI

[21] [21] Tomita, K. and Yamamuro, K., Lower bounds for fundamental units of real quadratic fields. Nagoya Math. J. 166(2002), 29–37. Google Scholar | DOI

[22] [22] Washington, L. C., A family of cyclic quartic fields arising from modular curves. Math. Comp. 57(1991), no. 196, 763–775. Google Scholar | DOI

[23] [23] Williams, H. C. and Zarnke, C. R., Computer calculation of units in cubic fields. Proceedings of the Second Manitoba Conference on Numerical Mathematics, Congressus Numerantium VII (1972), 433–468. Google Scholar

[24] [24] Wu, Q. and Scheidler, R., An explicit treatment of biquadratic function fields. Contrib. Discrete Math. 2(2007), no. 1, 43–60. Google Scholar

[25] [25] Zhang, Z.and Yue, Q., Fundamental units of real quadratic fields of odd class number. J. Number Theory 137(2014), 122–129. Google Scholar | DOI