This course is an introduction to Class Field
Theory, which is the study of abelian extensions of number fields. These
extensions are described in terms of arithmetic invariants such as the ideal
and ray class groups. One of the main results is Artin's
Reciprocity Law, which generalizes quadratic reciprocity, and can be viewed analytically
as a first case of Langlands Program. For more
information, see the Syllabus
and detailed Lecture Schedule.
Course Information
|
Scheduled Time |
Room |
|
Lectures |
TTh 10:30 |
Lockett 119 |
|
Office Hours |
T 4:30 |
Lockett 320 |
|
Textbook(s) |
N. Childress, Class Field Theory, Springer-Verlag, 2009. Available as an ebook through LSU
Libraries.
|
Note that the three primary sources are available electronically.
This section will list other resources for supplemental reading.
Textbooks and Lecture Notes:
Papers: (links provided to original sources; subscription or LSU campus-access
may be required)
Links to other resources:
|
Date |
Lecture Topics / Reading /
Handouts |
|
Tues.,
Aug. 27 |
Introduction to local
obstructions for Diophantine Equations; Definition of Legendre symbol and
Quadratic Reciprocity. |
|
Thurs.,
Aug. 29 |
Quadratic Reciprocity and
local solutions to quadratic equations; Properties of Cyclotomic
Fields (Ash Chap. 7, Ireland-Rosen 13.2); Algebraic Number Theory for Galois
Extensions (Ash 8.1) |
|
Tues.,
Sep. 3 |
Classical approach to
Quadratic Reciprocity via Algebraic Number Theory (Ireland-Rosen 13.3);
Galois theoretic approach to Quadratic Reciprocity (Ash 8.3) |
|
Thurs.,
Sep. 5 |
Dedekind-Kummer Theorem and prime factorization in cyclotomic fields, including characterization of prime
splitting (Ash 8.3, K. Conrad’s notes);
Frobenius automorphisms
and cyclotomic fields; Definition of Decomposition
Group and Orbit-Stabilizer Theorem (Ash 8.1) |
|
Tues.,
Sep. 10 |
Definition of local image
of Decomposition Group and Inertia Group (Ash 8.1); Statement of “Layer
Theorem” (Ash 8.2, Childress 1.1); Inertia Group and relative degree |
|
Thurs.,
Sep. 12 |
Order of Inertia Group (Ash
8.1); Examples: imaginary quadratic field, cyclotomic
field with trivial Inertia Group, and with nontrivial Inertia Group (see T.
Weston’s notes
for discussion of a larger cyclotomic field) |
|
Tues.,
Sep. 17 |
Frobenius automorphisms:
as generator of decomposition group; conjugation properties; Artin symbol; power relation for intermediate field
extensions, and restriction relation for Galois extensions (Ash 8.2) |
|
Thurs.,
Sep. 19 |
Dirichlet’s theorem for primes in arithmetic
progressions; characterization of splitting of primes in cyclotomic
extensions using Frobenius automorphisms
(Childress 1.1); discussion of Chebotarev Density
Theorem; Characters of finite abelian groups, canonical isomorphisms
(Childress 2.1) |
|
Tues.,
Sep. 24 |
Orthogonality relations for
characters, and orthogonality for roots of unity; indicator functions as
character sums (Childress 2.1); Dirichlet charcters, definition of conductor, induced characters,
primitive characters, real/quadratic characters (Childress 2.2) |
|
Thurs.,
Sep. 26 |
Kernels of character
groups, and fields associated to characters; Character groups and extended
Galois correspondence; Example of Conductor-Discriminant formula for cyclotomic fields; Character groups and factorization of
primes (Childress 2.2) |
|
Tues.,
Oct. 1 |
Calculation of
Decomposition and Inertia groups using characters (Childress 2.2); Example of
cyclotomic field; Definition of Dirichlet
series (Childress 2.3) |
|
Thurs.,
Oct. 3 |
Dirichlet L-functions; Abscissa of
convergence and absolute convergence; Euler products (Childress 2.3); Sketch
of proof of Dirichlet’s theorem for primes in
arithmetic progressions; Definition and basic properties of Dedekind zeta
functions (Childress 2.4) |
|
Tues.,
Oct. 8 |
Proof of non-vanishing of
L-functions for non-principal characters; Dirichlet
density (Childress 2.4); Density of set of splitting primes for Galois
extensions (Childress 2.5) |
|
Thurs.,
Oct. 10 |
Definition of Ray Class
Groups; Narrow ray class groups; Relation to class group (Childress 3.2; Table of class numbers for
imaginary quadratic fields) |
|
Tues.,
Oct. 15 |
Generalized Dirichlet Characters; Size of the ray class group;
Decomposition of ray class group into class group and field extension, via
Diamond Isomorphism theorem for groups (Childress 3.2) |
|
Tues.,
Oct. 22 |
Size of the ray class
group, continued; Examples of ray class group for real quadratic fields; Definition
of Ray Class Field; Definition of Congruence Subgroups; Uniqueness of Class
Field; Definition of Residue Classes (Childress 3.2). |
|
Thurs.,
Oct. 24 |
Proof of density for prime
residue classes; Relation between index of congruence subgroup, splitting
primes, and degree of field extension; Relative norm vs. Absolute norm;
Universal Norm Index Inequality (Childress 3.2) |
|
Tues.,
Oct. 29 |
Definition of Hilbert Class
Field; Example over quadratic number field (Childress 3.2); Absolute values
on number fields, a.k.a. places; Product Formula; Finite vs. Infinite places
(Childress 4.1) |
|
Thurs.,
Oct. 31 |
Topological Groups;
Homeomorphisms and continuity; Review of non-archimedean
metric topology; Examples and exploration of topology of Z_p
(Childress 4.2) |
|
Tues.,
Nov. 5 |
Compactness of Z_p, via equivalence with sequential compactness for
metric spaces (Childress 4.2, J. Hunter’s notes);
Restricted topological products; Definition of Ideles
and idelic units; Map of ideles
to fractional ideals; Diagonal
embedding (Childress 4.3) |
|
Thurs.,
Nov. 7 |
Ideal Class Group as
quotient of ideles; Discussion of algebraic closure
and completion for non-archimedean norms;
Definition of content map; Examples of places in number field extensions;
Sketch of proof of finiteness of class group (Childress 4.3; K. Conrad’s notes;
T. Weston’s article) |
|
Tues.,
Nov. 12 |
Ray Class Groups as
quotients of ideles; Ray classes and idele classes; Discussion of Takagi’s proof/construction
of class fields; Definition of idele norm
(Childress 4.3); Galois action on ideles (Childress
4.5) |
|
Thurs.,
Nov. 14 |
Properties of idelic norm and preparation for definition of conductor
(Childress 4.4 and 4.5); Definition of the conductor of a number field
(Childress 5.1) |
|
Tues.,
Nov. 19 |
Unramified primes and conductor, and surjectivity of norm (See Fesenko
–Vostokov Chap. 4 link);
Definition of Artin map and symbol; Statement of “Consistency”
theorem for Artin symbol (Childress 5.1) |
|
Thurs.,
Nov. 21 |
Proof of consistency; Useful
corollaries of consistency (Childress 5.1); Statement of Artin
Reciprocity (Lenstra-Stevenhagen’s article);
Example of ramified prime in quadratic number field, and relation to
quadratic reciprocity (Childress 5.2) |
|
Tues.,
Nov. 26 |
Proof of Artin Reciprocity for cyclotomic
number fields (Childress 5.2) |
|
Tues.,
Dec. 3 |
Sketch of proof of Artin Reciprocity for cyclic extensions (Childress 5.2);
Recovery of quadratic reciprocity as special case of Artin
Reciprocity (Childress 5.3) |
|
Thurs.,
Dec. 5 |
Sketch of proof of the
existence of class fields, including Ordering Theorem and Reduction Lemma
(Childress 6.1), and Kummer Extensions (Childress
6.2) |
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