Park City Mathematics Institute
Algebraic Number Theory
- Reflections from the Algebraic Number Theory SSTP Working Group
- Lori Bodner, Cliff Cheng, Brian Hopkins, Zack Korzyk, Armando Madrigal, David Metzler, Christopher Romero, Blue Taylor, Todd Vawdrey
- The Algebraic Number Theory working group attended the 1pm course taught by David Pollack for the Undergraduate Faculty Program. At 2pm each day, we debriefed and worked on problem sets. In addition, we had six hours of evening problem sessions attended by almost all group members.
- The course was motivated by asking what integers can be written in the form x2 + dy2 for integers x and y and a given integer d. This quickly led to exploration of number rings such as Z[i], the Gaussian integers, and Z[√-5], the collection of numbers having the form a + b√-5 for integers a and b. In some of these settings, familiar properties of the integers no longer hold. For instance, in Z[√-5], there are two factorizations of 6 into elements that do not factor further: 2*3 = (1 + √-5)*(1 - √-5). In order to "fix" this, we followed Dedekind's development of ideals. In the example, the ideal generated by 6 does factor into the "prime ideal" product (2, 1 + √-5)2 * (3, 1 + √-5) * (3, 1 - √-5). Pollack's presentation was noteworthy for several reasons: he gave several examples, introduced new machinery only after establishing the need for it, did not always prove statements in full generality, and repeatedly returned to the motivating question of writing integers in the form x2 + dy2.
- Each participant wrote a brief reflection on the experience, summarized below. Every participant explained that they enjoyed this mathematically challenging experience. Although the course content seemed to have no direct application to mathematics addressed in secondary schools, some participants found helpful parallels.
"As a high school teacher, I can see that posing a seemingly 'simple question' can have some profound mathematical consequences that are enjoyable to explore for their own sake and lead to other profound results."
"Working with ideals has been challenging but rewarding. It has helped me gain a better understanding of the conceptual difficulty our students face when encountering new number systems, such as irrational
or complex numbers. At the same time, it has reminded me of the creative aspect of mathematics, as developing a new concept enables a solution to an otherwise unsolved problem."
- There were many comments on the pedagogy of lecturing.
"I appreciated the contrast in the presentations (teacher-centered versus student-centered) between morning and afternoon. I understand that each course had a specific audience and I was able to contextualize the pros and cons of each approach based on audience."
"I have really enjoyed being in a classroom setting again. It has made me reflect on my own teaching practice. ... it has re-opened my eyes to how ineffective passive learning really is. ... This class has really solidified my determination to make my students active learners."
- Finally, there were comments on some of the practices we adapted in our working group sessions.
"The opportunity to work in a group on such challenging material has helped me to realize the importance of group work for my students' success. It has also allowed me the invaluable benefit of feeling what a good functioning group is like from the inside. I have realized also how frustrating note taking can be for students when they can't keep up with what's on the board. Working in the more formal setting of taking notes for an hour has also helped me to compare the experience first hand to the early morning class where we are learning through the discovery method. ... I've also appreciated the way that both Brian and David have given a bird's eye view of the topic by explaining the essence of a given lecture in a few sentences. This has helped me to keep everything in perspective and make connections that would have been much harder to make on my own. ... I hope to be able to apply what I've learned in my own classroom to make better use of group work, give a few sentences (or have the students do it) to capture the essence of every lesson both at the beginning and the end of class and to keep in mind how note giving and note taking should feel."
- In addition to the work of group facilitator Brian Hopkins, the expertise of participant David Metzler helped the group very much. We would also like to acknowledge the full participation of UFP member Patty McKenna in our 2pm and evening sessions.
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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. 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 Science Foundation.