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Class numbers of real quadratic function fields. (English) Zbl 0871.11081

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 \(h_M=h\).

MSC:

11R58 Arithmetic theory of algebraic function fields
11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
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