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Identical coursesAlgebraic Number Theory (B-KUL-G0A99A)
This course is taught this academic year, but not next year.Aims
This course introduces the basic concepts of algebraic number theory, which were developed at the turn of the nineteenth century in order to attack certain diophantine problems (i.e. to find the sets of integer or rational solutions to certain polynomial equations). The most celebrated success of this theory is Kummer's proof of Fermat's Last Theorem for regular exponents. Kummer's proof serves as a red line throughout the course, although various other types of diophantine problems are addressed:
- A classification of all prime numbers p that can be written as x2 + ny2 (for certain given values of n).
- Lagrange's theorem that every positive integer can be written as a sum of four squares.
- ...
Next, algebraic number theory has paved the road for many new branches of mathematics (such as class field theory, and even modern algebraic geometry) that surpass the original diophantine motivation by far. A secondary aim is to lift some tips of the veil here.
By the end of the course, the student should be able to attack various kinds of diophantine problems using the techniques of algebraic number theory. He / she should have a thorough understanding of the underlying theory, and of its range of applicability (e.g. why does Kummer's proof fail for non-regular exponents?).
Previous knowledge
Knowledge of algebra, as for example provided in the courses Algebra I (G0N88B) and Algebra II (G0P53A), is necessary. Students taking Algebra II and Algebraic Number Theory in the same semester will have to read on Galois Theory in the course notes of Algebra II before it is treated in class.
Basic knowledge of number theory, as for example provided in the course Number Theory (G0P61B), is recommended.
Knowledge of commutative algebra, as for example provided in the course Commutative Algebra (G0A82A), can be helpful, but is not essential.
Is included in these courses of study
Activities
5 ects. Algebraic Number Theory (B-KUL-G0A99a)
Content
- Cultural background: history of Fermat's Last Theorem, Fermat's proof of the case n = 4, Lamé's erroneous proof
- Update on commutative algebra: norms, traces, discriminants, Dedekind domains, unique ideal factorization, class groups and class numbers
- Number fields and rings of integers: quadratic numbers fields, integral bases, ramification indices and degrees, norms of ideals
- Geometric representation: Minkowski's lemma + applications, geometric representations, logarithmic representations
- Finiteness theorems: finiteness of the class number, Dirichlet's unit theorem, Dedekind's theorem on ramification, Hermite's theorem
- Connections with Galois theory: Frobenius elements, decomposition and intertia groups, Chebotarev's density theorem (without proof) + applications
- Cyclotomic fields: cyclotomic polynomials, primes in arithmetic progressions, Fermat's Last Theorem for regular exponents
Course material
Course notes + Toledo.
1 ects. Algebraic Number Theory: Exercises (B-KUL-G0B02a)
Content
See G0A99a.
Course material
Exercise sets + Toledo.