# REU Computational Research on Local Fields and Galois Groups

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## Projects

We describe some of the possible research projects for this REU. We don not expect that you have even heard of the mathematical objects on this page.- Generating polynomials for special cases
- Galois groups of polynomials in special cases
- Ramification polygons of higher order
- Possible automorphism groups of polynomials with given invariants
- Possible Galois groups of polynomials with given invariants
- Database of local fields
- Generalization of known results to the equal characteristic case

### Past Projects

We give a collection of student research projects on local fields and Galois groups directed by the organizers of the REU.

* denotes an undergraduate student

** denotes a graduate student

- Constructing Galois 2-extensions of the 2-adic numbers C. Awtrey with J. Beuerle, and J. Schrader*
- Subfields of solvable sextic field extensions C. Awtrey with P. Jakes*
- Efficient computation of Galois groups of even sextic polynomials C. Awtrey with P. Jakes*, submitted. Test polynomials can be accessed HERE.
- Determining Galois groups of reducible polynomials via discriminants and linear resolvents C. Awtrey with T. Cesarski*, and P. Jakes*, JP J. Algebra, Num. Theory Appl.,
**39**, no. 5, 685-702,*2017*. - Computing Galois groups of Eisenstein polynomials over p-adic fields Jonathan Milstead** directed by S. Pauli, PhD Thesis, UNCG 2017
- Symbolic Computation of Resolvents Sandi Rudzinski** directed by S. Pauli, MA Thesis, UNCG 2017
- Enumerating Extensions of (π)-Adic Fields with Given Invariants S. Pauli with Brian Sinclair**. Tables of Number of extensions of local fields
- Algorithms for computing quartic Galois groups over fields of characteristic 0 C. Awtrey with J. Beuerle, and M. Keenan*, Int. J. Pure Appl. Math.,
**112**, no. 4, 709-740,*2017*. - Galois groups of degree 12 2-adic fields with trivial automorphism group C. Awtrey with B. Barkley*, N. Miles*, C. Shill*, and E. Strosnider*, JP J. Algebra, Num. Theory Appl.,
**38**, no. 5, 457-471,*2016*. - Degree 12 2-adic fields with automorphism group of order 4 C. Awtrey with B. Barkley*, N. Miles*, C. Shill*, and E. Strosnider*, Rocky Mountain J. Math.,
**45**, no. 6, 1755-1764,*2016*. - Constructing Splitting Fields of Polynomials over Local Fields S. Pauli with Jonathan Milstead** and Brian Sinclair**
- Irreducible sextic polynomials and their absolute resolvents C. Awtrey with R. French*, P. Jakes*, and A. Russell, Minn. J. Undergraduate Math.,
**1**, no. 1, 14-32,*2015*. - Algorithms for Enumerating Invariants and Extensions of Local Fields Brian Sinclair** directed by Sebastian Pauli, PhD Thesis, UNCG, 2015
- Centralizers of transitive permutation groups and applications to Galois theory C. Awtrey with N. Mistry* and N. Soltz*, Missouri J. Math. Sci.,
**27**, no. 1, 16-32,*2015* - On Galois group of degree 15 polynomials C. Awtrey with K. Mazur, S. Rodgers*, N. Soltz*, and J. Weed*, Int. J. Pure Appl. Math.,
**104**, no. 3, 407-420,*2015*. - Degree 14 2-adic fields C. Awtrey with N. Miles*, J. Milstead**, C. Shill*, and E. Strosnider*, Involve,
**8**, no. 2, 329-336,*2015*. - Groups of order 16 as Galois groups over the 2-adic numbers C. Awtrey with J. Johnson*, J. Milstead**, and B. Sinclair**, Int. J. Pure Appl. Math.,
**103**, no. 4, 781-795,*2015*. - Absolute resolvents and masses of irreducible quintic polynomials C. Awtrey with C. Shill*, Collaborative Mathematics and Statistics Research: Topics from the 9th Annual UNCG Regional Mathematics in Statistics Conference, Springer Proceedings of Mathematics & Statistics, Springer, New York,
**109**, 31-41,*2015*. - A linear resolvent for degree 14 polynomials C. Awtrey with E. Strosnider*, Collaborative Mathematics and Statistics Research: Topics from the 9th Annual UNCG Regional Mathematics in Statistics Conference, Springer Proceedings of Mathematics & Statistics, Springer, New York,
**109**, 43-50,*2015*. - Resolvents, masses, and Galois groups of irreducible quartic polynomials C. Awtrey with B. Barkley*, M. McCraw*, and J. Guinn*, Pi Mu Epsilon Journal,
**13**, no. 10, 609-618,*2014*. - Galois groups of degree 12 2-adic fields with automorphism group of order 6 or 12 C. Awtrey with C. Shill*, Topics from the 8th Annual UNCG Regional Mathematics and Statistics Conference, Springer Proceedings in Mathematics & Statistics, Springer, New York,
**64**, 55-65,*2013*. - Dihedral p-adic fields of prime degree C. Awtrey with T. Edwards*, Int. J. Pure Appl. Math.,
**75**, 185-194,*2012*.

Chad Awtrey, Department of Mathematics and Statistics,
Elon University

Sebastian Pauli, Number Theory Group
Department of Mathematics and Statistics,
UNCG