18.786 (Topics in Algebraic Number Theory) Calendar

Dates in italics do not indicate class days. Note that proofs were omitted for most topics in class field theory (i.e., starting May 2), but most of the references below provide them.

References: Janusz = Algebraic Number Fields; Neukirch = Algebraic Number Theory (translated from the German); Cassels-Fröhlich = Algebraic Number Theory (out of print); Milne's notes = Class Field Theory, available at James Milne's web site. For class field theory, see also my Math 254B course notes (Berkeley, spring 2002).




February 6 (M)


February 7 (T)

Course overview

addition on elliptic curves

February 9 (R)

Localization, examples; integral dependence, integral closure; discrete valuation rings (definition)

Janusz I.1-3

February 14 (T)

Discrete valuation rings (properties), Dedekind domains, unique factorization of ideals

Janusz I.3

February 16 (R)

Fractional ideals of a Dedekind domain, class group, finite extensions of fields, norm, trace, discriminant

Janusz I.4-5

February 21 (T)


February 23 (R; KSK away)


February 28 (T)

Trace and norm, separability, nondegeneracy of the trace pairing for a separable extension, extension of Dedekind domains in the separable case

Janusz I.5-6

March 2 (R)

Extension of prime ideals, relative degree, ramification degree, the fundamental equality, discriminant

Janusz I.6-7

March 7 (T)

Discriminants and ramification, norms of ideals

Janusz I.7-8

March 9 (R)

Norm of a prime ideal; properties of cyclotomic fields (prime power case)

Janusz I.8, I.10

March 14 (T)

Linearly disjoint extensions; cyclotomic fields (general case)

Janusz I.9, I.10, I.11; see also this supplement

March 16 (R)

Why quadratic reciprocity is now easy; real and complex embeddings, lattices

Janusz I.11, I.12, I.13

March 21 (T)

Lattices and ideal classes, Minkowski's theorem, finiteness of the class group; Dirichlet's units theorem

Janusz I.12, I.13

March 23 (R)

Proof of Dirichlet's units theorem

Janusz I.13

March 27-31

NO CLASS (MIT spring break)

April 4 (T)

Absolute values;completions of fields with respect to an absolute value, examples; dichotomy between archimedean nonarchimedean absolute values; absolute values coming from discrete valuation rings; normalized absolute values (places), statement of the product formula for number fields; classification of completions of the rational numbers (Ostrowski's theorem)

Janusz II.1-II.3

April 6 (R)


April 11 (T)

Ostrowski's theorem continued; exponential and logarithm series; Hensel's lemma for nonarchimedean absolute values; extensions of nonarchimedean absolute values

Janusz II.2, II.3

April 13 (R)

Extension of nonarchimedean absolute values

Janusz II.3

April 18 (T)

NO CLASS (Patriots Day)

April 20 (R)

Classification of absolute values on a number field; product formula for number fields; unramified extensions

Janusz II.3, II.5

April 25 (T)

Decomposition and inertia groups, Frobenius elements, Artin symbols

Janusz III.1, III.2

April 27 (R)

Artin maps for abelian extensions; ray class groups; the Artin reciprocity law; proof in the cyclotomic case

Janusz III.3, IV.1

May 2 (T)

More on ray class groups; idelic interpretation

Janusz IV.1; Neukirch VI.1

May 4 (R)

Dirichlet series, Dedekind zeta functions, L-series, Dirichlet's theorem and generalizations

Janusz IV.2

May 9 (T)

Chebotarev density theorem; Arakelov class group (notes by Rene Schoof)

Janusz IV.3

May 11 (R)

Arakelov class group; local class field theory

Schoof notes; Neukirch III, Milne's notes

May 16 (T)

Local class field theory continued; the adelic reciprocity map; the principal ideal theorem

Neukirch III and VI, Milne's notes, Cassels-Fröhlich

May 18 (R)

Class field towers; complex multiplication; TAKE-HOME FINAL DUE