text: The Fontaine-Mazur conjectures ask whether l-adic representations of these mathematical objects have convenient geometric properties.
full_question: The Fontaine-Mazur conjectures ask whether l-adic representations of these mathematical objects have convenient geometric properties. Infinite groups of this type can be endowed with the Krull topology. The "absolute" one for a field F is defined in terms of the separable closure of F. For field L and subfield K, all automorphisms of L fixing K are trivial if and only if the corresponding one of these groups is trivial, since the order of one of these groups equals the degree of the corresponding field. A polynomial equation is solvable by radicals if and only if the one of these groups corresponding to its splitting field is solvable. For 10 points, name these groups of automorphisms of field extensions, whose subgroups correspond to intermediate fields according to the namesake "theory" of a French mathematician.
first_sentence: The Fontaine-Mazur conjectures ask whether l-adic representations of these mathematical objects have convenient geometric properties.
raw_answer: Galois groups [prompt on "groups" or "automorphism groups"]