Jun, Lee, and Sun posed the question of whether the cyclotomic Hecke field can be generated by a single critical L-value of a cyclotomic Hecke character over a totally real field. They provided an answer to this question in the case where the tame Hecke character is trivial. In this presentation, we explain an extension of their work to address the case of non-trivial Hecke characters over solvable totally real number fields. Our approach builds upon the primary estimation obtained by Jun-Lee-Sun, supplemented with new inputs, including global class field theory, duality principles, the analytic behavior of partial Hecke L-functions, and the non-vanishing of twisted sum of Gauss sums and Hyper Kloosterman sums. This is a Zoom seminar. Please join us at: https://cnrs.zoom.us/j/95421205116?pwd=h09AKYQPbXmECgcDFam7WYoWzMoDuD.1