proposition: Lemma minCpoly_aut nu x : minCpoly (nu x) = minCpoly x.
proof: Proof. wlog suffices dvd_nu: nu x / (minCpoly x %| minCpoly (nu x))%R. apply/eqP; rewrite -eqp_monic ?minCpoly_monic //; apply/andP; split=> //. by rewrite -{2}(algC_autK nu x) dvd_nu. have [[q [Dq _] min_q] [q1 [Dq1 _] _]] := (minCpolyP x, minCpolyP (nu x)). rewrite Dq Dq1 dvdp_map -min_q -(fmorph_root nu) -map_poly_comp. by rewrite (eq_map_poly (fmorph_rat nu)) -Dq1 root_minCpoly. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun ssrnat eqtype seq choice) mathcomp(div fintype path bigop finset prime order ssralg) mathcomp(poly polydiv mxpoly generic_quotient countalg) mathcomp(ssrnum closed_field ssrint archimedean rat intdiv) mathcomp(algebraics_fundamentals)