Friesen, Christian; van Wamelen, Paul Class numbers of real quadratic function fields. (English) Zbl 0871.11081 Acta Arith. 81, No. 1, 45-55 (1997). Let \(\mathbb{F}_q\) denote the finite field of \(q\) elements with odd characteristic. If \(M\in \mathbb{F}_q[t]\) is a degree 4, monic irreducible then we write \(h_M\) as the ideal class number of the ‘real’ quadratic extension \(\mathbb{F}_q(t,\sqrt{M(t)})\). We prove two main results. First we show that, for all \(q\) (where \(\text{char}(\mathbb{F}_q)\geq 5)\), there exist at least \(\frac{q^{7/2}}{10\log\log q}\) monic irreducible quartics \(M\in \mathbb{F}_q[t]\) such that \(h_M=1\). In addition we determine that for any odd positive \(h\), for all sufficiently large \(q\) (where \(\text{char}(\mathbb{F}_q)\geq 5)\), there exists an irreducible monic quartic \(M\in \mathbb{F}_q[t]\) with \(h_M=h\). Reviewer: C.Friesen (Marion, OH) Cited in 1 ReviewCited in 3 Documents MSC: 11R58 Arithmetic theory of algebraic function fields 11R29 Class numbers, class groups, discriminants 11R11 Quadratic extensions Keywords:class numbers; functions fields; elliptic curves; Gauss conjecture; monic irreducible quartics PDFBibTeX XMLCite \textit{C. Friesen} and \textit{P. van Wamelen}, Acta Arith. 81, No. 1, 45--55 (1997; Zbl 0871.11081) Full Text: DOI EuDML