This is the repo for the book Unreal Analysis, written by Paul Pollack and published by the Ross Mathematics Program.

Click here for a PDF of the manuscript, and here to purchase a physical copy.

To report typos and other errors, please open an issue.

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How to use this book

This manuscript is intended for undergraduates and beginning graduate students looking to get started with non-Archimedean concepts.

For instructors using this text in a class on $p$-adic numbers: Congratulations! You are likely aware that courses on this topic at this level are rare. If used as the primary course text, problems should be distributed to students without solutions. Class time can then be used for discussing ideas and approaches, led by students but facilitated by the instructor. Experts will often see more in the problems than beginners, and discussions can be used to share these insights.

Solutions to Problem Set ($p$-Set) X should be distributed once the class is ready to move on to Set X+1. For semester systems, covering one set per week is a reasonable goal, with the understanding that later sets may take more time. This book can also be used as a supplementary resource, with the instructor choosing problems to enhance their own exercise sheets.

For students: Congratulations! $p$-adic numbers are a (un)real treat to think about. If you are using this book for a course, follow your instructor's directions. If using it for self-study, I recommend tackling the problems systematically, completing one set before moving on to the next. Give yourself enough time to think through each problem before seeking solutions. If certain problems prove too difficult, solutions are provided for that purpose.

I must emphasize that this should not be your only resource on $p$-adic numbers. The author learned about them from other sources! After mastering each problem set, I encourage you to compare the content with that in the suggested readings. True understanding comes from seeing a topic from all angles, and different texts highlight different perspectives.

Suggestions for further reading

• Cassels, J. W. S. (1986). Local fields. Cambridge University Press.
• Conrad, K. Expository papers. https://kconrad.math.uconn.edu/blurbs/
• Gouvêa, F. Q. (2020). $p$-adic numbers: An introduction (3rd ed.). Springer.
• Katok, S. (2007). $p$-adic analysis compared with real. American Mathematical Society.
• Koblitz, N. (1984). $p$-adic numbers, $p$-adic analysis, and zeta-functions (2nd ed.). Springer.
• Murty, M. R. (2002). Introduction to $p$-adic analytic number theory. American Mathematical Society.
• Schikhof, W. H. (2006). Ultrametric calculus: An introduction to $p$-adic analysis. Cambridge University Press.
• Serre, J.-P. (1973). A course in arithmetic. Springer.
• Steuding, J. Die $p$-adischen Zahlen. https://www.uni-marburg.de/de/fb12/fachbereich/profil/geschichte-des-fachbereichs/biographisches/hensel_p_adischen_zahlen.pdf