REU Computational Research on Local Fields and Galois Groups

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In the local field \(\mathbb{Q}_5\):
  1. \(\sqrt{-1}\in\mathbb{Q}_5\)
  2. \(\mathbb{Q}_5\) has infinitely many algebraic extensions
  3. \(4+4\cdot 5+4\cdot 25+\dots = -1\)
  4. \(|1|_5 > |5|_5\)
  5. \(\sum_{n=1}^\infty a_n\) converges \(\Leftrightarrow\) \(\lim_{n\to\infty}a_n=0\)
Figure 1: Local fields are more fun than the real numbers.

A Research Experience for Undergraduates (REU) for 8 students on Local Fields and Galois Groups from June 4 to July 27 will take place at Elon University in North Carolina. Participants will earn a $4000 stipend and will receive: travel support to and from Elon University, free lodging in Elon dorms, and free lunch on weekdays. The deadline to apply for the 2018 REU is Sunday March 15, for details see Application.

Research Topics

The topic of Local Fields and Galois Groups is an area of pure mathematics that is well suited for undergraduate research. The explicit presentation of local fields make them accessible to undergraduate students, and their applications in number theory and other areas of mathematics make results of the undergraduate research applicable in the research of others. An important example of local fields are the p-adic fields.

In the local ring \(\mathbb{Z}_5\):
  1. \(\frac{1}{2}\in \mathbb{Z}_5\)
  2. \(\mathbb{Z}_5\) contains only one prime
  3. \(\mathbb{Z}_5\) is complete (with respect to \(|\cdot|_5\))
  4. the multiplicative group \(\mathbb{Z}_5^\times\) is generated by \(\sqrt{-1}\) and \(1+5\)
  5. \(|a|_5\le 1\) for all \(a\in \mathbb{Z}_5\)
Figure 2: Local rings are more fun than the integers.

In the projects we will follow constructive apporach to local fields and Galois theory. All projects from this REU will prepare students to make research contributions that extend the frontiers of research, including broad dissemination of results in the form of conference presentations and publications journals. On the Projects page we give a list of possible projects for the REU and some examples of past projects directed by the organizers.

Environment

The participants will take part in Elon University's campus-wide undergraduate research program Our REU participants will join this community of scholars during the summer, staying in the same dormitories, participating in the same community-wide social events, and disseminating their results at a joint end-of-the-summer Presentation Symposium.

Acknowledgements NSA logo

The REU Computational Research on Local Fields and Galois Groups is supported by Elon University and the NSA.


Chad Awtrey, Department of Mathematics and Statistics, Elon University
Sebastian Pauli, Number Theory Group Department of Mathematics and Statistics, UNCG