REU Computational Research on Local Fields and Galois Groups
ProjectsWe 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
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.