Congruence function fields with class number one (Q2823075)

\textit{P. Mercuri} and \textit{C. Stirpe} [``Classification of algebraic function fields with class number one'', Preprint, \url{arXiv:1406.5365}] showed that, up to isomorphism, there is only one function field \(K\) with exact field of constants \(\mathbb F_2\) of genus \(g_K=4\) and with class number \(h_K=1\). In the paper under review, the authors provide a different proof of this result. They use the Carlitz-Hayes theory of cyclotomic function fields, which may be outlined as follows: Let \(k=\mathbb F_q(T)\), with \(k^{ac}\) the algebraic closure of \(k\). For \(u\in k^{ac}, M\in\mathbb F_q[T]\), define \(\Phi(M)\in \text{End}_{\mathbb F_q}k^{ac}\) by \(\Phi(M)(u)=M(\phi+\mu)(u)\), where \(\phi:k^{ac}\rightarrow k^{ac}\), \(\phi(u)=u^q\) is the Frobenius automorphism and \(\mu:k^{ac}\rightarrow k^{ac}, \mu(u)=Tu\) is multiplication by \(T\). Then \(\Phi\) endows \(k^{ac}\) with an \(\mathbb F_q[T]\)-module structure, and is called a Carlitz module. If \(\Lambda_M\) denotes the \(M\)-torsion points of \(k^{ac}\) under this action, then the field \(k(\Lambda_M)\) (obtained by adjoining the points of \(\Lambda_M\) to \(k\)) is called a cyclotomic function field (associated with \(M\)). To establish the main result, the authors first show that if \(K\) is a function field over \(\mathbb F_2\) such that \(g_K=4\) and \(h_K=1\), then \(K\) has a unique rational subfield \(k=\mathbb F_2(T)\) such that \([K:k]=5\) and \(K/k\) is a cyclic extension. They then use a result of \textit{M. L. Madan} and \textit{C. S. Queen} [Acta Arith. 20, 423--432 (1972; Zbl 0237.12007)] to deduce that, up to isomorphism, necessarily \(K\subseteq k(\Lambda_M)\mathbb F_{2^5}\), where \(M=T^4+T+1\). The bulk of the proof is contained in Theorem 2.5, which asserts that, up to isomorphism, there exists only one field \(K\) such that \(k\subseteq K\subseteq k(\Lambda_M)\mathbb F_{2^5}\), and the main result (Theorem 2.7) then follows easily from this.