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.

## Dates

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

• 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

#DateContentNotes
128.10.1. What this is all about
1.1. The sum of two squares problem
1.2. Review
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230.10.2. Field extensions
2.1 Constructing field extensions
2.2. Minimal polynomials and algebraic elements
2.3 Splitting fields
34.11.2.4 Conjugate roots and separability (continued)
2.5 Primitive elements
2.6 Characteristic polynomial, norm, trace
2.7 Trace form and discriminant
PDF
46.11.2.7 Trace form and discriminant (continued)
3. Rings of integers
3.1 Integral elements
3.2 Modules (review)
3.3 Integral elements form a ring
PDF
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)
PDF
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
PDF
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
PDF
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)
PDF
1027.11.5.6 The round-2 algorithm (theory) (continued)
5.7 Computing in orders
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112.12.5.9 Computing the p-multiplier
6. Geometry of numbers (Minkowski theory)
6.1 Lattices
6.2 Lattices in Euclidean space
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124.12.6.3 Quadratic supplement and Cholesky decomposition
6.4 Minkowski theory
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139.12.6.5 Discreteness of lattices
6.6 Shortest vectors and lattice density
PDF
1411.12.6.6 Shortest vectors and lattice density (continued)
6.7 Successive minima
6.8 Lattice reduction
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1516.12.6.8 Lattice reduction (continued)
7. Units
7.1 Torsion units
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1618.12.7.2 Units are finitely generated
7.3 Free rank of the unit group
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176.1.7.3 Free rank of the unit group (continued)
7.4 Remarks
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188.1.8. Ideal theory of rings of integers
8.1 Fractional ideals
8.2 Dedekind domains
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1913.1.8.2 Dedekind domains (continued)
8.3 Finiteness of the class group
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2015.1.8.3 Finiteness of the class group (continued)
8.4 Ramification theory
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2120.1.8.4 Ramification theory (continued)
8.5 Geometric interlude
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2222.1.8.5 Geometric interlude (continued)
8.6 Computing factorizations
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2327.1.Lecture by Claus Fieker
2429.1.Lecture by Claus Fieker
253.2.8.7 (Un)ramified primes
8.8 Galois theory of primes
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265.2.8.8 Galois theory of primes (continued)
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2710.2.8.10 Ramification in cyclotomic fields
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## Software

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.

## Literature

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)

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)