This course is an introduction to algebraic number theory (number fields, units, splitting of primes, class groups, etc.). In addition to theory, we will also discuss basic algorithms that allow (in principle) to explicitly compute most of the objects we will consider.


Lectures are on Mondays and Wednesdays, both at 13:45–15:15 in room 48–438. The first lecture is on Oct 28. Exercise sessions are on Tuesdays, 8:15–9:45 in room 48–438.

Exercise sheets

1 (28.10.), 2 (4.11.), 3 (11.11.), 4 (18.11.), 5 (25.11.), 6 (02.12.), 7 (09.12.), 8 (16.12), 9 (6.1.), 10 (13.1.), 11 (20.1.), 12 (3.2.)

Additional material

  • I turned Exercise 4.4(c) about the Hermite and Smith normal forms of a matrix over the Gaussian integers into a little computer algebra project. There, you'll also find the solution to the exercise.
  • Here's the complete solution to Exercise 5.1(a), where we compute the maximal order in L(α) for α a root of X3-X2-2X-8 using the round-2 algorithm.
  • Here's my (brutal) Magma program that computes units with the Dirichlet unit theorem (Exercise 9.3)
  • Solution to Exercise 11.3.
  • Solutions to Exercise Sheet 12.

Lecture notes

128.10.1. What this is all about
1.1. The sum of two squares problem
1.2. Review
230.10.2. Field extensions
2.1 Constructing field extensions
2.2. Minimal polynomials and algebraic elements
2.3 Splitting fields Conjugate roots and separability (continued)
2.5 Primitive elements
2.6 Characteristic polynomial, norm, trace
2.7 Trace form and discriminant
PDF Trace form and discriminant (continued)
3. Rings of integers
3.1 Integral elements
3.2 Modules (review)
3.3 Integral elements form a ring
511.11.3.4 Ring of integers is integrally closed
3.5 Integrality of minimal polynomial, norm, trace
3.6 Ring of integers is finitely generated
3.7 Ring of integers is free (including review of free modules)
613.11.3.7 Ring of integers is free (continued, including structure of modules over PIDs)PDF
718.11.3.7 Ring of integers is free (continued)
4. Hermite and Smith normal form
820.11.4. Hermite and Smith normal form (continued)
5. Finding an integral basis
5.1 Orders, discriminants, a sufficient condition
5.2 Discriminant of an equation order
5.3 Zassenhaus approach
925.11.5.3 Zassenhaus approach (continued)
5.4 Review of prime ideals and radicals
5.5 Primes in an order
5.6 The round-2 algorithm (theory)
1027.11.5.6 The round-2 algorithm (theory) (continued)
5.7 Computing in orders
5.8 Computing the p-radical
PDF Computing the p-multiplier
5.10 Round-2 made constructive
6. Geometry of numbers (Minkowski theory)
6.1 Lattices
6.2 Lattices in Euclidean space
PDF Quadratic supplement and Cholesky decomposition
6.4 Minkowski theory
PDF Discreteness of lattices
6.6 Shortest vectors and lattice density
1411.12.6.6 Shortest vectors and lattice density (continued)
6.7 Successive minima
6.8 Lattice reduction
1516.12.6.8 Lattice reduction (continued)
7. Units
7.1 Torsion units
1618.12.7.2 Units are finitely generated
7.3 Free rank of the unit group
PDF Free rank of the unit group (continued)
7.4 Remarks
188.1.8. Ideal theory of rings of integers
8.1 Fractional ideals
8.2 Dedekind domains
1913.1.8.2 Dedekind domains (continued)
8.3 Finiteness of the class group
2015.1.8.3 Finiteness of the class group (continued)
8.4 Ramification theory
2120.1.8.4 Ramification theory (continued)
8.5 Geometric interlude
2222.1.8.5 Geometric interlude (continued)
8.6 Computing factorizations
2327.1.Lecture by Claus Fieker
2429.1.Lecture by Claus Fieker (Un)ramified primes
8.8 Galois theory of primes
PDF Galois theory of primes (continued)
8.9 Ramification in quadratic extensions
2710.2.8.10 Ramification in cyclotomic fields
8.11 Quadratic reciprocity


Computer algebra systems supporting number theory: Hecke (developed in Kaiserslautern!), PARI/GP, Sage, Magma. You can also try to develop algorithms from scratch in, e.g., Julia or Python.


  1. J. Milne has excellent notes on Fields and Galois theory and on Algebraic Number Theory
  2. H. Cohen, A Course in Computational Algebraic Number Theory (Springer GTM, 1996)
  3. M. Pohst, H. Zassenhaus, Algorithmic Algebraic Number Theory (Cambridge University Press, 1989)
  4. J. Neukirch, Algebraic Number Theory (Springer, 1999)

Reading recommendations

  1. H.M. Edwards, The Background of Kummer's Proof of Fermat's Last Theorem for Regular Primes (1975)
  2. S. Gelbart, An elementary introduction to the Langlands program (1984)
  3. D. Wübben, D. Seethaler, J. Jaldén, and G. Matz, Lattice reduction—A survey with applications in wireless communications (2011)