MATH 7230 - Fall 2018

Algebraic Number Theory


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



MWF 1:30

Lockett 134

Office Hours

M 2:30

Lockett 320




Robert Ash, A course in Algebraic Number Theory, Dover Publications, 2010.

J. S. Milne, Algebraic Number Theory, online lecture notes.

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


Problem Sets


Other references

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:


Lecture Schedule



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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