An algebraic number field is a field obtained by adjoining to the rational numbers the roots of an irreducible rational polynomial. Algebraic number theory is the study of properies of such fields. This course will cover the basics of algebraic number theory, with topics to be studied possibly including the following: number fields, rings of integers, factorization in Dedekind domains, class numbers and class groups, units in rings of integers, valuations and local fields, and zeta- and L-functions.
Prof. Paul Gunnells, LGRT 1115L, 545–6009, gunnells at umass dot edu. The best way to contact me is by email. Please don’t leave a message on my office phone or through any other way; I won’t get it.
TBA. Office hours will be held by Zoom. An invitation will be sent to the email address listed in SPIRE.
Algebraic Number Theory by James Milne
Other valuable references:
Cassels, Frohlich, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965). A standard reference, with expositions of many topics, including local/global fields, cohomology of groups, class field theory, towers of class fields, Hecke L-functions and their functional equations.
Borevich, A. I.; Shafarevich, I. R. Number theory. Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20 Academic Press, New York-London 1966. A classic. Very hands on. Lots of examples.
Manin, Yuri Ivanovic; Panchishkin, Alexei A. Introduction to modern number theory. Fundamental problems, ideas and theories. Translated from the Russian. Second edition. Encyclopaedia of Mathematical Sciences, 49. Springer-Verlag, Berlin, 2005. Many precise definitions, but few complete proofs. Gives a wide overview of the subject.
Koch, H. Algebraic number theory. Translated from the 1988 Russian edition. Reprint of the 1992 translation. Springer-Verlag, Berlin, 1997. The companion volume to Manin-Panchishkin. Similar in style. An excellent way to see the whole subject in the large without getting bogged down in the details.
Frohlich, A.; Taylor, M. J. Algebraic number theory. Cambridge Studies in Advanced Mathematics, 27. Cambridge University Press, Cambridge, 1993. Emphasizes the role of valuations.
Neukirch, Jürgen Algebraic number theory. Translated from the 1992 German original and with a note by Norbert Schappacher. With a foreword by G. Harder. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322. Springer-Verlag, Berlin, 1999. Excellent book. Lots of material, including class field theory. Nice treatment of number fields from the point of view of Arakelov geometry. Zeta and L-functions too (the approach complements the material in Cassels-Frohlich). Full details of just about everything.
Lang, Serge Algebraic number theory. Second edition. Graduate Texts in Mathematics, 110. Springer-Verlag, New York, 1994. A classic. Unique tone.
Marcus, Number Fields. Springer-Verlag, New York, 1977. Another classic. Very hands on, with many detailed examples and exercises. Highly recommended by certain number theorists for self-study.
Rodriguez Villegas, Fernando. Experimental number theory. Oxford Graduate Texts in Mathematics, 13. Oxford University Press, Oxford, 2007. Shows how to use computation to explore number theory and to formulate conjectures.
The grading for the course will be as follows.
50% will be based on homework problems. Homework will be assigned during the term. Instructions for submitting problems will be given once the grader is chosen. You may work on problems in groups, but it is expected that you will submit your own writeups of solutions. It is also expected that your solutions will be original work.
50% will be based on class participation. Attendance is not taken in the course. Nevertheless I expect you will be an active paricipant in the course by attending lectures, asking questions, etc.
The University of Massachusetts Amherst is committed to providing an equal educational opportunity for all students. If you have a documented physical, psychological, or learning disability on file with Disability Services (DS), you may be eligible for reasonable academic accommodations to help you succeed in this course. If you have a documented disability that requires an accommodation, please notify me within the first two weeks of the semester so that we may make appropriate arrangements.
Since the integrity of the academic enterprise of any institution of higher education requires honesty in scholarship and research, academic honesty is required of all students at the University of Massachusetts Amherst. Academic dishonesty is prohibited in all programs of the University. Academic dishonesty includes but is not limited to: cheating, fabrication, plagiarism, and facilitating dishonesty. Appropriate sanctions may be imposed on any student who has committed an act of academic dishonesty. Instructors should take reasonable steps to address academic misconduct. Any person who has reason to believe that a student has committed academic dishonesty should bring such information to the attention of the appropriate course instructor as soon as possible. Instances of academic dishonesty not related to a specific course should be brought to the attention of the appropriate department Head or Chair. Since students are expected to be familiar with this policy and the commonly accepted standards of academic integrity, ignorance of such standards is not normally sufficient evidence of lack of intent. For more information see the website of Dean of Students Office.
Expectations for our course as as follows:
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