MATH 7230 - Fall 2019

Class Field Theory


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



TTh 10:30

Lockett 119

Office Hours

T 4:30

Lockett 320




N. Childress, Class Field Theory, Springer-Verlag, 2009.

Available as an ebook through LSU Libraries.

J. S. Milne, Class Field Theory, online lecture notes.

W. Stein, Algebraic Number Theory, A Computational Approach, online lecture notes.


Problem Sets


Other references

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:


Lecture Schedule



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 FesenkoVostokov 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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