# IMAGINARY BICYCLIC FUNCTION FIELDS WITH THE REAL CYCLIC SUBFIELD OF CLASS NUMBER ONE

• Jung, Hwan-Yup (Department of Mathematics Education Chungbuk National University)
• Published : 2008.05.31

#### Abstract

Let $k={\mathbb{F}}_q(T)$ and ${\mathbb{A}}={\mathbb{F}}_q[T]$. Fix a prime divisor ${\ell}$ q-1. In this paper, we consider a ${\ell}$-cyclic real function field $k(\sqrt[{\ell}]P)$ as a subfield of the imaginary bicyclic function field K = $k(\sqrt[{\ell}]P,\;(\sqrt[{\ell}]{-Q})$, which is a composite field of $k(\sqrt[{\ell}]P)$ wit a ${\ell}$-cyclic totally imaginary function field $k(\sqrt[{\ell}]{-Q})$ of class number one. und give various conditions for the class number of $k(\sqrt[{\ell}]{P})$ to be one by using invariants of the relatively cyclic unramified extensions $K/F_i$ over ${\ell}$-cyclic totally imaginary function field $F_i=k(\sqrt[{\ell}]{-P^iQ})$ for $1{\leq}i{\leq}{\ell}-1$.

#### References

1. J. Ahn, S. Bae, and H. Jung, Cyclotomic units and Stickelberger ideals of global function fields, Trans. Amer. Math. Soc. 355 (2003), no. 5, 1803-1818 https://doi.org/10.1090/S0002-9947-03-03245-8
2. S. Bae and J. Koo, Genus theory for function fields, J. Austral. Math. Soc. Ser. A 60 (1996), no. 3, 301-310 https://doi.org/10.1017/S1446788700037824
3. V. Fleckinger and C. Thiebaud, Ideaux ambiges dans les corps de fonctions, J. Number Theory 100 (2003), no. 2, 217-228 https://doi.org/10.1016/S0022-314X(02)00102-6
4. P.-L. Kang and D.-S. Lee, Genus numbers and ambiguous class numbers of function fields, Commun. Korean Math. Soc. 12 (1997), no. 1, 37-43
5. M. Rosen, Ambiguous divisor classes in function fields, J. Number Theory 9 (1977), no. 2, 160-174 https://doi.org/10.1016/0022-314X(77)90018-X
6. H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, Berlin, 1993
7. C. Wittmann, l-class groups of cyclic function fields of degree l, Finite Fields Appl. 13 (2007), no. 2, 327-347 https://doi.org/10.1016/j.ffa.2005.09.001
8. H. Yokoi, On the class number of a relatively cyclic number field, Nagoya Math. J. 29 (1967), 31-44 https://doi.org/10.1017/S0027763000024119
9. H. Yokoi, Imaginary bicyclic biquadratic fields with the real quadratic subfield of class-number one, Nagoya Math. J. 102 (1986), 91-100 https://doi.org/10.1017/S0027763000000441
10. J. Zhao, Class number relation between type (l,l,...,l) function fields over $F_q(T)$ and their subfields, Sci. China Ser. A 38 (1995), no. 6, 674-682

#### Cited by

1. HILBERT 2-CLASS FIELD TOWERS OF REAL QUADRATIC FUNCTION FIELDS vol.29, pp.2, 2014, https://doi.org/10.4134/CKMS.2014.29.2.219