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)


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