## Galois groups with prescribed ramification.(English)Zbl 0815.11053

Childress, Nancy (ed.) et al., Arithmetic geometry. Conference on arithmetic geometry with an emphasis on Iwasawa theory, March 15-18, 1993, Arizona State Univ., Tempe, AZ, USA. Providence, RI: American Mathematical Society. Contemp. Math. 174, 35-60 (1994).
Let $$K$$ be an algebraic number field or a function field in one variable $$x$$. The author considers normal extensions of $$K$$ with prescribed ramification. The background for this talk is the conjecture of Abhyankar, proved by the author, that in the case of an algebraically closed constant field of characteristic $$p > 0$$ and $$n$$ ramified places one can realize all finite groups $$G$$ as Galois groups over $$K$$ with the following property: Let $$p(G)$$ be the normal subgroup of $$G$$ generated by the $$p$$-Sylow groups of $$G$$. Then $$G/p (G)$$ has to have not more than $$2g + n - 1$$ generators, where $$g$$ denotes the genus of $$K$$ [Invent. Math. (to appear)].
In section 1 the author considers the case of a function field with algebraically closed or finite constant field. In section 2 he considers number fields mostly in connection with analogs of Abhyankar’s conjecture. There are many interesting examples.
For the entire collection see [Zbl 0802.00017].
Reviewer: H.Koch (Berlin)

### MSC:

 11R32 Galois theory 14H30 Coverings of curves, fundamental group 12F12 Inverse Galois theory 12F10 Separable extensions, Galois theory