The Minimal Ramification Problem for Rational Function Fields over Finite Fields.

The article investigates the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois groups, focusing on analogues of conjectures known for number fields. It formulates Conjecture 1.4 predicting that for a prime power \( q = p^\nu \) and a finite group \( G \), the minimal number of ramified primes in geometric Galois extensions of \( \mathbb{F}_q(T) \) with Galois group \( G \) equals \(\max\{ d((G/p(G))^{\mathrm{ab}}), 1 \}\), where \( p(G) \) is the subgroup generated by the \( p \)-Sylow subgroups and \( d(\cdot) \) denotes the minimal number of generators. The authors prove this conjecture for abelian groups and establish upper bounds and exact values for symmetric groups \( S_n \) and alternating groups \( A_n \) in many cases, including products of such groups, using explicit polynomial constructions, ramification theory, and group-theoretic classification results. They also relate these function field results to the classical minimal ramification problem over \( \mathbb{Q} \), showing conditionally (under Schinzel's hypothesis H) that \( r_{\mathbb{Q}}(S_n) = 1 \) for all \( n \). The work employs tools from Galois theory, finite group theory (including the classification of finite simple groups), and analytic number theory over function fields, providing new insights into ramification minimality in positive characteristic and its analogies with number fields.