Equality of orders of a set of integers modulo a prime

For finitely generated subgroups W 1 , … t W_1, \ldots , W_t of alttext="double-struck Q Superscript times"> Q ×<!-- × </mml:msup> encoding="application/x-tex">\mathbb {Q}^{\times } , integers alttext="k k k encoding="application/x-tex">k_1, k_t a Galois extension F"> F encoding="application/x-tex">F Q"> {Q} and union conjugacy classes C subset-of normal G l left-parenthesis F slash double-struck right-parenthesis"> C ⊂<!-- ⊂ mathvariant="normal">G mathvariant="normal">a mathvariant="normal">l ( / stretchy="false">) encoding="application/x-tex">C \subset \mathrm {Gal}(F/\mathbb {Q}) we develop methods for determining if there exist infinitely many primes alttext="p"> p encoding="application/x-tex">p such that the index reduction i"> i encoding="application/x-tex">W_i modulo divides encoding="application/x-tex">k_i Artin symbol on is contained in C"> encoding="application/x-tex">C . The results are multivariable generalization H.W. Lenstra’s work. We present several applications, including characterization all alttext="a n a n encoding="application/x-tex">a_1, a_n alttext="normal o r d p Baseline right-parenthesis equals mathvariant="normal">o mathvariant="normal">r mathvariant="normal">d = encoding="application/x-tex">\mathrm {ord}_p(a_1) = {ord}_p(a_n) obtained conditional to Riemann hypothesis.