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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 9 students on Local Fields and Galois Groups took place at Elon University in North Carolina from June 4 to July 27, 2018.

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 follow constructive approach to local fields and Galois theory. All projects from this REU 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.

Environment

The participants take part in Elon University's campus-wide undergraduate research program Our REU participants 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

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

Disclaimer

Any opinions, findings, and conclusions or recommendations, expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Security Agency.

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