On D 5 -polynomials with integer coefficients
Kishi, Yasuhiro1
1 Department of Mathematics Fukuoka University of Education 1-1 Bunkyoumachi Akama, Munakata-shi Fukuoka, 811-4192 Japan
Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 113-125.

We give a family of D 5 -polynomials with integer coefficients whose splitting fields over ℚ are unramified cyclic quintic extensions of quadratic fields. Our polynomials are constructed by using Fibonacci, Lucas numbers and units of certain cyclic quartic fields.

DOI : 10.5802/ambp.258
Classification : 11R29
Mots-clés : class number, Fibonacci number, polynomial
Kishi, Yasuhiro 1

1 Department of Mathematics Fukuoka University of Education 1-1 Bunkyoumachi Akama, Munakata-shi Fukuoka, 811-4192 Japan 
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Kishi, Yasuhiro. On $D_5$-polynomials with integer coefficients. Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 113-125. doi : 10.5802/ambp.258. http://www.numdam.org/articles/10.5802/ambp.258/

[1] Ankeny, N. C.; Chowla, S. On the divisibility of the class number of quadratic fields, Pacific J. Math., Volume 5 (1955), pp. 321-324 | MR | Zbl

[2] Byeon, Dongho Real quadratic fields with class number divisible by 5 or 7, Manuscripta Math., Volume 120 (2006) no. 2, pp. 211-215 | DOI | MR | Zbl

[3] Ichimura, H. Note on the class numbers of certain real quadratic fields, Abh. Math. Sem. Univ. Hamburg, Volume 73 (2003), pp. 281-288 | DOI | MR | Zbl

[4] Imaoka, Masafumi; Kishi, Yasuhiro On dihedral extensions and Frobenius extensions, Galois theory and modular forms (Dev. Math.), Volume 11, Kluwer Acad. Publ., Boston, MA, 2004, pp. 195-220 | MR | Zbl

[5] Mestre, Jean-François Courbes elliptiques et groupes de classes d’idéaux de certains corps quadratiques, J. Reine Angew. Math., Volume 343 (1983), pp. 23-35 | DOI | MR | Zbl

[6] Nagell, T. Über die Klassenzahl imaginär-quadratischer Zahlköper, Abh. Math. Sem. Univ. Hamburg, Volume 1 (1922), pp. 140-150 | DOI

[7] Nakamura, S. A microcosm of Fibonacci numbers (Japanese), Nippon Hyoronsha Co., Tokyo, 2002

[8] Parry, Charles J. On the class number of relative quadratic fields, Math. Comp., Volume 32 (1978) no. 144, pp. 1261-1270 | DOI | MR | Zbl

[9] Ribenboim, Paulo The new book of prime number records, Springer-Verlag, New York, 1996 | MR | Zbl

[10] Sase, Masahiko On a family of quadratic fields whose class numbers are divisible by five, Proc. Japan Acad. Ser. A Math. Sci., Volume 74 (1998) no. 7, pp. 120-123 | DOI | MR | Zbl

[11] Weinberger, P. J. Real quadratic fields with class numbers divisible by n, J. Number Theory, Volume 5 (1973), pp. 237-241 | DOI | MR | Zbl

[12] Yamamoto, Yoshihiko On unramified Galois extensions of quadratic number fields, Osaka J. Math., Volume 7 (1970), pp. 57-76 | MR | Zbl

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