Galois extensions ramified at one prime

This thesis studies Galois extensions of global fields and associated Galois groups with one ramified prime, in both the number field and function field cases. Over [special characters omitted] some restrictions on both solvable and nonsolvable Galois groups ramified only at one prime are shown. We also give a description of tamely ramified meta-abelian Galois groups and examples of infinite class field towers with one finite prime ramified. Over real quadratic fields [special characters omitted], results about nilpotent Galois groups ramified only at one prime in [special characters omitted] are shown. Over function fields [special characters omitted]q(t), a stronger version of the forward direction of “Abhyankar\u27s conjecture” is proved. Finally, we give a simple description of cyclotomic function fields with a view toward developing Iwasawa theory for cyclotomic function fields