We classify extensions of function fields of curves over finite fields in which the class number does not change. This breaks up into three parts, of which we will emphasize the third in this talk. 1. Identification of a finite set of possible pairs of Weil polynomials for the two curves, and of cyclic covers consistent with this set by way of explicit class field theory (presented at ANTS-XV in August 2022). 2. Proof using 1 that the only extensions that can occur are cyclic (presented at AGC^2T in June 2023). 3. Identification of curves of genus up to 7 whose Weil polynomials are candidates for the base of the extension. This uses Mukai's explicit description of the universal curves over the various Brill-Noether strata of the moduli stacks of curves of genus 6 and 7, with some attention paid to working over a nonclosed (here finite) base field.