 

Moshe Jarden and
Nantsoina Cynthia Ramiharimanana
Embedding problems with bounded ramification over function fields of positive characteristic
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print


Published: 
December 8, 2020. 
Keywords: 
embedding problems, bounded ramification, function fields of positive characteristic. 
Subject: 
12F12, 12E30. 


Abstract
Let K_{0} be an algebraic function field of one variable over a Hilbertian field F of positive
characteristic p. Let K be a finite Galois extension of K_{0}. We prove that every finite embedding problem
1>H>G>Gal(K/K_{0})>1 whose kernel H is a pgroup is properly solvable.
Moreover, the solution can be chosen to locally coincide with finitely many, given in advance, weak local solutions.
Finally, and this is the main point of this work, the number of prime divisors of K_{0}/F that ramify in the solution field is bounded by the number of prime divisors of K_{0} that ramify in K plus the length of the maximal Ginvariant sequence of subgroups of H. 

Acknowledgements
The authors are indebted to Aharon Razon for carefully reading drafts of this work and for his useful comments. Special thanks go to the anonymous referee who suggested an essential simplification of an earlier version of the proof of Proposition 6.8.


Author information
Moshe Jarden:
School of Mathematical Sciences
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University
Ramat Aviv, Tel Aviv 69978, Israel
jarden@tauex.tau.ac.il
Nantsoina Cynthia Ramiharimanana:
Department of Mathematics
Clemson University
Clemson, SC 29634, USA
nantsoina@aims.ac.za

