This course is an introduction to algebraic
number theory, the goal of which is to determine the structure of
multiplication in rings (particularly those that extend the integers). We will
cover the standard results, culminating in Dirichlet's
Unit Theorem, and then use local rings to develop the alternative adelic approach. See the Syllabus and detailed Lecture Schedule.
Course Information
|
Scheduled Time |
Room |
|
Lectures |
MWF 1:30 |
Lockett 134 |
|
Office Hours |
M 2:30 |
Lockett 320 |
|
Textbook(s) |
Robert Ash, A course in
Algebraic Number Theory, Dover Publications, 2010. |
Note that the three primary sources are available as online
lecture notes. 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 |
|
Mon.,
Aug. 20 |
Fundamental Theorem of
Arithmetic; Unique factorization in the Gaussian integers and Eisenstein integers;
Integer lattices in the complex plane |
|
Wed.,
Aug. 22 |
Euclidean algorithm for
Gaussian integers; Primes that are representable as the sum of two squares
(Ash Chapter 1); Legendre and Lagrange's three and four-square theorems |
|
Fri.,
Aug. 24 |
Algebraic and integral
elements (Ash 1.1); Integral ring extensions characterized as modules |
|
Mon.,
Aug. 27 |
Calculating monic
polynomials for integral elements using Cramer's Rule (Ash 1.1); Fifth/tenth
roots of unity and the Golden Ratio (J. Peters’ paper) |
|
Wed.,
Aug. 29 |
Classification of algebraic
integers in quadratic number fields (Ash 2.2); Integral closure; Fraction
fields (Ash 1.1) |
|
Fri.,
Aug. 31 |
Rings of fractions; Localization
of rings at prime ideals; Properties of ideals in rings of fractions (Ash
1.2); The case of PIDs |
|
Wed.,
Sep. 5 |
Characterization of prime
ideals in rings of fractions; Local rings and localization at prime ideals
(Ash 1.2) |
|
Fri.,
Sep. 7 |
Discussion of local rings
and p-adic rings/fields; Traces, norms, and
characteristic polynomials of field extensions (Ash 2.1) |
|
Mon.,
Sep. 10 |
Properties of traces and
norms; Relation between characteristic and minimal polynomials; Calculation
of traces and norms from minimal polynomial (Ash 2.1) |
|
Wed.,
Sep. 12 |
Separable field extensions
and Galois automorphisms (Ash 2.1); Dedekind’s
Lemma (K. Conrad’s notes) |
|
Fri.,
Sep. 14 |
AKLB: General setup for
Algebraic Number Theory; Basis of algebraic elements for field extensions
(Ash 2.2) |
|
Mon.,
Sep. 17 |
Discriminant in a field
extension; Detection of linear independence; Relation to Galois automorphisms; Discriminant of power basis and Van der
Monde determinant; Relation to polynomial discriminant (Ash 2.3) |
|
Wed.,
Sep. 19 |
Calculation of cubic
discriminant; Algebraic integers over a PID; Invariance of discriminant (Ash
2.3) |
|
Fri.,
Sep. 21 |
Field discriminant and squarefree case (Ash 2.3); Basic properties of Noetherian
rings and Dedekind domains (Ash 3.1, K. Conrad’s notes)
|
|
Mon.,
Sep. 24 |
Integral closure of
Dedekind domains (Ash 3.1); Fractional ideals in Dedekind domains;
Multiplication of fractional ideals (Ash 3.2) |
|
Wed.,
Sep. 26 |
Construction of inverses of
prime ideals; Multiplicative group of fractional ideals (Ash 3.2, B. Conrad’s
notes) |
|
Fri.,
Sep. 28 |
Unique factorization of
(fractional) ideals; Equivalence of unique factorization and principal ideals
for Dedekind domains (Ash 3.3, B. Conrad’s notes);
Divisibility and containment of ideals (Ash 3.3) |
|
Mon.,
Oct. 1 |
Definition of Ideal Class
Group; Equivalence of trivial class group and PID (Ash 3.4); Lifting prime
ideals in field extensions; Factorization into prime ideals lying over base
prime; Ramification and relative degree; Examples of prime factorization in
quadratic number fields (Ash 4.1) |
|
Wed.,
Oct. 3 |
Examples of prime
factorization in a cyclotomic field; The “efg”-relation (Ash 4.1); Case of Galois Field Extension
(Ash 8.1) |
|
Mon.,
Oct. 8 |
Norms of Ideals; Examples
of ideal factorization; Relation between norms of ideals and discriminants
(Ash 4.2) |
|
Wed.,
Oct. 10 |
Multiplicativity of ideal norms; Relation
to relative index; Finiteness of ideals of bounded norm (Ash 4.2); Examples
of ideal factorization |
|
Fri.,
Oct. 12 |
Ramification of primes and
discriminants (Ash 4.2) |
|
Mon.,
Oct. 15 |
Dedekind/Kummer Factorization Algorithm for rings of algebraic
integers with power bases; Classification of prime ideal factorization for
quadratic number fields and quadratic reciprocity (Ash 4.3) |
|
Wed.,
Oct. 17 |
Lattices, fundamental
domain and volume, Pigeonhole principle and lattice points (Ash 5.1) |
|
Fri.,
Oct. 19 |
Minkowski’s Convex Body Theorem (Ash
5.1); Cylinders and Volume calculation for the Canonical Embedding (Ash 5.2) |
|
Mon.,
Oct. 22 |
Canonical Embedding for
rings of algebraic integers; Relation between fundamental volume,
discriminant, and norms of ideals (Ash 5.3) |
|
Wed.,
Oct. 24 |
Fundamental embedding for
proper ideals, including examples and review of Smith Normal Form (Ash 5.3; ) |
|
Fri.,
Oct. 26 |
Proof of Minkowski’s Bound of Ideal Norms; Finiteness of Class
Number; Calculation of Ideal Class Group (Ash 5.3) |
|
Mon.,
Oct. 29 |
Logarithmic Embedding and
lattices; Structure of roots of unity in number fields (Ash 6.1) |
|
Wed.,
Oct. 31 |
Structure of units in
number fields; Discrete Subgroups (Ash 6.1); Dirichlet’s
Unit Theorem (Ash 6.2) |
|
Fri.,
Nov. 2 |
Proof of Dirichlet’s Unit Theorem, using Minkowski’s
Convex Body Theorem; Fundamental System of Units (Ash 6.2); Examples of units
in quadratic fields, including Pell’s equation and efficient solutions using
periodic continued fractions (Ash 6.3) |
|
Mon.,
Nov. 5 |
Absolute Values and Discrete
Valuations, and associated norms/metric spaces; Non-archimedean
absolute values; p-adic valuation (Ash 9.1) |
|
Wed.,
Nov. 7 |
Induced absolute values,
including p-adic norm; Valuation Ring and Valuation
Ideal, particularly as local rings (Ash 9.1) |
|
Fri.,
Nov. 9 |
Discrete Valuation Rings;
Triangles with respect to a nonarchimedean metric
(Ash 9.1); Absolute values on the rationals and Ostrowski’s Theorem (Ash 9.2) |
|
Mon.,
Nov. 12 |
Proof of Ostrowski’s Theorem (B. Conrad’s survey);
Product Formula for absolute values (Ash 9.2); Artin-Whaples
Approximation Theorem, including motivation from Chinese Remainder Theorem
(Ash 9.3; Artin and Whaples’ paper) |
|
Wed.,
Nov. 14 |
Proof of Artin-Whaples Theorem; Example of approximation with
respect to multiple valuations in rationals (Ash
9.3) |
|
Fri.,
Nov. 16 |
Cauchy sequences and null
sequences; Completions of fields (Ash 9.4) |
|
Mon.,
Nov. 19 |
Cauchy sequences vs.
Dedekind Cuts; Diagonal Embedding; Proof of completeness; Nonarchimedean
completions and local fields, including Residue Fields (Ash 9.4) |
|
Mon.,
Nov. 26 |
Laurent series expansion in
local fields; Projective/inverse limits; p-adic
field and p-adic integers (Ash 9.4; Calculator for p-adic square roots; For many examples, see J. Cremona’s
lecture notes) |
|
Wed.,
Nov. 28 |
Local-Global principle (K.
Conrad’s survey);
Hasse-Minkowski Theorem for rational solutions to
homogeneous quadratic forms; Chevalley-Warning
Theorem, proof of special case (B. Mazur’s lecture notes);
Selmer’s cubic counterexample to Local-Global principle (E. Selmer’s paper;
M. Bhargava’s paper); Hensel’s
Lemma (Ash 9.5) |
|
Fri.,
Nov. 30 |
Proof of Hensel’s Lemma
(Ash 9.5); Introduction to Global Fields and Adeles,
including extensions of absolute values, infinite places, ring of adeles, group of ideles and
1-ideles, and proof of finiteness of Class Number (Stein Chap. 17-19) |
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