Computing Galois groups of Eisenstein polynomials over p-adic fields

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Jonathan Milstead (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site:
Sebastian Pauli

Abstract: The most efficient algorithms for computing Galois groups of polynomials over global fields are based on Stauduhar’s relative resolvent method. These methods are not directly generalizable to the local field case, since they require a field that contains the global field in which all roots of the polynomial can be approximated. We present splitting field-independent methods for computing the Galois group of an Eisenstein polynomial over a p-adic field. Our approach is to combine information from different disciplines. We primarily, make use of the ramification polygon of the polynomial, which is the Newton polygon of a related polynomial. This allows us to quickly calculate several invariants that serve to reduce the number of possible Galois groups. Algorithms by Greve and Pauli very efficiently return the Galois group of polynomials where the ramification polygon consists of one segment as well as information about the subfields of the stem field. Second, we look at the factorization of linear absolute resolvents to further narrow the pool of possible groups.

Additional Information

Language: English
Date: 2017
Galois, Hensel, Resultant, Rudzinski, Sinclair, Yasaki
Galois theory
p-adic fields

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