MATH 5020 – Infinite Galois Theory

 

Course Description:

Description: Infinite Galois Theory

Description: This is a second course in Galois theory. We will begin with a review of the fundamental theorem of Galois theory that is taught in Abstract Algebra II and discuss a number of explicit examples in zero and positive characteristic (to explore separability issues), including finite fields. The course will emphasize both theoretical and computational methods (with software such as Sage and Magma) to study field extensions. Correspondences similar to the Galois correspondence show up in other areas of mathematics, such as in complex analysis and topology, and we will emphasize these similarities. The main goal is to discuss Galois theory of infinite extensions (esp. of finite fields, the rational numbers, and function fields). The usual Galois correspondence breaks down for infinite extensions, but can be saved by introducing topological ideas: infinite Galois groups are best viewed not as abstract groups, but as compact topological groups. Once again, we will work with explicit examples such as the (separable) closure of a finite field, or cyclotomic extensions. Time permitting, we will introduce Galois representations, and produce a few examples.

Prerequisites: This is a graduate level abstract algebra class, so familiarity with algebra (at the undergraduate level) is expected.

Meets: at MONT 420, on Tuesdays and Thursdays, from 2:00pm – 3:15pm.


About the Instructor: Álvaro Lozano-Robledo

Alvaro Lozano-Robledo in action (Shawn Kornegay/UConn Photo)
Alvaro Lozano-Robledo in action (Shawn Kornegay/UConn Photo), see this UConn Today’s article.

I’m an associate professor of mathematics, and the director of the Quantitative Learning Center at the University of Connecticut

Office hours: Tuesdays 11-12, Wednesdays 10-11, or by appointment, at MONT 233.


Enrollment:

The enrollment for this class is by permission for undergraduate students. Send me a message if you would like to enroll.


Outline, books, and other resources:

The course will cover the following topics:

  1. Classical Galois Theory. We will review classical Galois theory using the book “Abstract Algebra“, by David S. Dummit, Richard M. Foote, 3rd edition (or 2nd edition, really). Other references for this section (and some below) include “Galois Theory“, by David Cox, and “Fields and Galois Theory“, by Milne (free!).
    1. Field extensions and automorphism groups, in characteristic 0 and characteristic p.
    2. The Fundamental Theorem of Galois Theory.
    3. Extensions of Finite fields.
    4. Composite extensions, simple extensions, the primitive element theorem.
    5. Cyclotomic extensions, and the Kronecker-Weber theorem.
    6. Galois groups of quadratic and cubic polynomials.
  2. Infinite Extensions.
    1. Algebraic closures. See this handout by Keith Conrad, and Ch. 6 of Milne’s “Fields and Galois Theory”.
    2. Cyclotomic extensions. See Washington’s “Introduction to Cyclotomic Fields”.
    3. Maximal 2-elementary extension.
  3. Profinite Groups.
    1. Inverse/projective limits. See Washington’s Appendix to “Introduction to Cyclotomic Fields”, and Lenstra’s notes.
  4. Topological Spaces and Topological Groups. We will draw from Munkres’ “Topology”, Hewitt/Ross “Abstract Harmonic Analysis”, Higgins’ “Introduction to Topological Groups“, and Pontryagin’s “Topological Groups”. See also these notes by Ryan Vinroot.
    1. Basic properties of topological groups.
    2. The Krull topology.
  5. The Fundamental Theorem of Infinite Galois Theory. Milne’s “Fields and Galois Theory”, Ch. 7.
    1. Proof.
    2. Examples.
  6. l-adic Galois Representations. Serre’s “Abelian l-adic Galois representations”.
    1. Definition.
    2. Examples.

Homework, Quizzes:

Homework will be assigned and collected every other week. Homework will be graded on completion only. If you have questions about how to solve a problem, you should talk to me in office hours or ask questions in class.

Homework: 

The homework problem sets will be posted here:

  1.  Set 1 (due September 14th).
    • Dummit and Foote: Section 14.1, #1, 5, 7; Section 14.2, #1, 3, 4, 5, 14, 15, 17, 18, 23, 26.
    • Find polynomials f over Q such that Gal(f) is isomorphic to each of the following groups: Z/2Z, Z/3Z, S3, Z/2Z x Z/2Z, Z/4Z, D8 (dihedral order 8), A4, S4.
    • Let F be the field with 3 elements. Find polynomials f over F such that Gal(f) is isomorphic to each of the groups above, or explain why such polynomial does not exist.
    • Let F be the field with 3 elements, and let K=F(T). Find polynomials f over K such that Gal(f) is isomorphic to each of the groups above, or explain why such polynomial does not exist.
  2. Set 2 (due September 28th).
    • Dummit and Foote: Section 14.3, #4, 6, 7, 8, 9; Section 14.4, #1, 2, 5, 6.
    • The following exercises are for students with an algebraic number theory background (reference: Marcus’ “Number Fields”, Chapter 4).
      • Let K be the splitting field of x^3-2, let G=Gal(K/Q), and let O be the ring of integers of K. Describe the factorizations into prime ideals of (2) = 2O, (3), (5), and (7). For each prime ideal P above p=2, 3, 5, and 7, describe the decomposition and inertia subgroups D=D(P/p) and I=I(P/p), explicitly as subgroups of G, and also determine K^D and K^I.
      • Let F be the 25-th cyclotomic field, let G=Gal(F/Q), and let O be the ring of integers of F. Describe the factorizations into prime ideals of (3) = 3O, (5), (7), and (11). For each prime ideal P above p= 3, 5, 7, and 11, describe the decomposition and inertia subgroups D=D(P/p) and I=I(P/p), explicitly as subgroups of G, and also determine F^D and F^I.
      • In each of the examples above, find a prime ideal P over p, such that the extension (O/P) / (Z/pZ) is non-trivial, and find an element of D(P/p) that maps to the Frobenius automorphism of Gal((O/P) / (Z/pZ)).
  3. Set 3 (due October 12th).
    • Dummit and Foote: Section 14.5, #7, 8, 9, 12; Section 14.6, #2, 4, 5, 6.
    • Lenstra’s notes: Exercises 1.3, 1.4, 1.11.
  4. Set 4 (due October 26).
    • Lenstra’s notes: Exercises 1.1, 1.2, 1.5, 1.6, 1.7, 1.8, 1.9, 1.10.

Class Grade:

The grade will be computed based on homework performance and class participation.

The total grade will be computed as follows:


Course Outline:

DF = Dummit and Foote
H = Higgins’ “An introduction to topological groups”
M = Milne’s “Field and Galois theory”
S = Serre’s “Abelian l-adic Galois representations”
W = Washington’s “Introduction to cyclotomic fields”

Week Topics Notes
1  Introduction: finite and infinite Galois theory
 Automorphisms of field extensions DF – 14.1
2  The fundamental theorem of Galois theory 1 DF – 14.2
 The fundamental theorem of Galois theory 2 DF – 14.2
3  Introduction to Magma. Finite Fields DF – 14.3, Magma calculator and handbook.
 Composite extensions and simple extensions DF – 14.4
4  Cyclotomic extensions, abelian extensions, and cyclotomic characters DF – 14.5
 Galois groups of quadratic, cubic, and quartic polynomials; Algebraic closures DF – 14.6
5  Algebraic closures; Infinite abelian extensions Constructing algebraic closures by Keith Conrad; W – Chapters 2 and 3.
 Infinite groups: p-adic integers, p-adic numbers; infinite products and profinite groups, inverse/projective limits Lenstra’s notes on profinite groups.
6  Projective limits of Galois groups; construction of other p-adic fields
 Topological groups, subgroups H – 2.1
7  Quotient groups H – 2.2
 Products H – 2.3
8  Systems of neighborhoods H – 2.4
 [Missed class]
9  Separation axioms, open subgroups H – 2.5
 Open subgroups H – 2.6
10  Connectedness H – 2.7
 Compactness  H – 2.8
11  Profinite groups  H – 2.9
 Proof of the Galois correspondence  M – Chapter 7
12  Proof of the Galois correspondence  M – Chapter 7
 Examples of the Galois correspondence (cyclotomic extensions)
13  Examples of the Galois correspondence (finite fields)
 Examples of the Galois correspondence (maximal 2-elementary abelian extension of Q)
14  Intro to splitting of prime ideals in extensions, decomposition groups, inertia groups
 The absolute Galois group of Q_p
15  l-adic Galois representations  S – 1.1
 Examples (with a brief introduction to elliptic curves, and etale cohomology representations)  S – 1.2

Policy Statements:

Please refer to http://provost.uconn.edu/syllabi-references/ for the common policies we follow at UConn.

  • Policy Against Discrimination, Harassment and Inappropriate Romantic Relationships — The University is committed to maintaining an environment free of discrimination or discriminatory harassment directed toward any person or group within its community – students, employees, or visitors.  Academic and professional excellence can flourish only when each member of our community is assured an atmosphere of mutual respect.  All members of the University community are responsible for the maintenance of an academic and work environment in which people are free to learn and work without fear of discrimination or discriminatory harassment.  In addition, inappropriate Romantic relationships can undermine the University’s mission when those in positions of authority abuse or appear to abuse their authority.  To that end, and in accordance with federal and state law, the University prohibits discrimination and discriminatory harassment, as well as inappropriate Romantic relationships, and such behavior will be met with appropriate disciplinary action, up to and including dismissal from the University. (More information is available at http://policy.uconn.edu/?p=2884.)
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  • Attendance — Your instructor expects you to attend class regularly. Besides being nearly essential for developing your understanding of the material, your regular attendance in class is good for the morale of the class and is indicative of your interest in the subject and your engagement in the course. You are responsible for the material discussed in class and in the assigned reading in the text.
  • Student Conduct Code — Students are expected to conduct themselves in accordance with UConn’s Student Conduct Code.
  • Academic Integrity Statement — This course expects all students to act in accordance with the Guidelines for Academic Integrity at the University of Connecticut. Because questions of intellectual property are important to the field of this course, we will discuss academic honesty as a topic and not just a policy. If you have questions about academic integrity or intellectual property, you should consult with your instructor. Additionally, consult UConn’s guidelines for academic integrity.
  • Students with Disabilities — The Center for Students with Disabilities (CSD) at UConn provides accommodations and services for qualified students with disabilities. If you have a documented disability for which you wish to request academic accommodations and have not contacted the CSD, please do so as soon as possible. The CSD is located in Wilbur Cross, Room 204 and can be reached at (860) 486-2020 or at csd@uconn.edu. Detailed information regarding the accommodations process is also available on their website at www.csd.uconn.edu.
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