Indivisibility of divisor class numbers of Kummer extensions over the rational function field
Let \(K\) be a Kummer extension over the rational function field \(k={\mathbb F}_q(T)\) of degree \(\ell\), where \({\mathbb F}_q\) is the finite field of order \(q\) and \(\ell\) is a prime divisor of \(q-1\). In Theorems 1.1 and 1.2 of the paper under review the authors obtain results regarding the indivisibility of the divisor class number \(h_K\) of \(K\) by \(\ell\) in two cases, respectively. In the first case they assume that the infinite prime of \(k\) is ramified in \(K\) and prove that \(h_K\) is not divisible by \(\ell\) if and only if there exists exactly one finite prime of \(k\) which is ramified in \(K\). Moreover, when \(h_K\) is not divisible by \(\ell\), then \(h_K\equiv 1\pmod \ell\). Equivalently, the divisor class number \(h_K\) of \(K\) is divisible by \(\ell\) if and only if there exists at least two finite primes of \(k\) which are ramified in \(K\). In the second case it is assumed that the infinite prime of \(k\) is unramified in \(K\). In this case the authors prove that \(h_K\) is not divisible by \(\ell\) if and only if the number of finite primes of \(k\) which are ramified in \(K\) is at most two and one of the following three cases holds: \begin{itemize}\item[(i)] \(K\) is an unramified extension over \(k\); \item[(ii)] If there is only one finite prime of \(k\) which is ramified in \(K\), then the genus of \(K\) is divisible by \(\ell\)\text; \item[(iii)] If there are exactly two finite primes \(P_1\) and \(P_2\) of \(k\), of degree \(d_1\) and \(d_2\), respectively, which are ramified in \(K\), then \(d_i\not\equiv 0 \pmod \ell\) for \(i=\text{1}\) or \(i=\text{2}\). \end{itemize} In all three cases (i), (ii), and (iii), we have \(h_K\equiv 1 \pmod \ell\). Using the above indivisibility criterion, the authors obtain in Theorem 1.3 an infinite family of maximal real cyclotomic function fields whose divisor class numbers are divisible by \(\ell\) where \(\ell\) is a prime divisor of \(q-1\). More precisely, for an irreducible monic polynomial \(P(T)\in {\mathbb F}_q[T]\) let \(k(\Lambda_{P})\) and \(k(\Lambda_{P})^+\) denote, respectively, the \(P\)-th cyclotomic function field and its maximal real subfield. Let \(d\) be an even integer such that \(q^{d/2}+1\) is squarefree and \(4\;|\;d\) if \(\ell=2\). Let \(P_0(T)\in {\mathbb F}_q[T]\) be an irreducible monic polynomial of degree \(d\) divisible by \(\ell\). Let \(\alpha\) be a root of \(P_0(T)\) in an algebraic closure \(\overline {\mathbb F}_q\) of \({\mathbb F}_q\) and let \(Q=q^d\). Then there exists a prime number \(m\) which satisfies the following: \begin{itemize}\item[(i)] \(\alpha \not \in ({\mathbb F}_Q)^m\)\text{;} \item[(ii)] \(m \;|\;Q-1\). \end{itemize} Now, for a nonnegative integer \(n\), let \(P_n(T)=P_0(T^{m^n})\), and let \(K_n=k(\sqrt[\ell]{P_n(T)})\) be a Kummer extension over \(k\) of degree \(\ell\). Then the divisor class number of \(K_n\) is divisible by \(\ell\) for each \(n\). Furthermore, for each \(n\), \(k(\Lambda_{P_n})^+\) has divisor class number divisible by \(\ell\). In the final section of the paper the authors present some examples illustrating their results.
scientific article; zbMATH DE number 6943098
scientific article