I will prepare lecture notes during the course of the semester.
The various drafts will appear here, with a date indicating which
version it is.
Version
of 18.12.2024.
The lecture will be accompanied by roughly biweekly exercise classes, usually during the Friday class. We will announce the precise dates in the lecture as well as here. You should submit your exercise sheets as a PDF upload to the SamUpTool.
| Dates of exercise classes |
|---|
| September 27 |
| October 11 |
| October 25 |
| November 8 |
| November 22 |
| December 13 |
| Exercise sheet | Due by | Solutions |
|---|---|---|
| Exercise sheet 1 | September 30 | Solutions 1 |
| Exercise sheet 2 | October 14 | Solutions 2 |
|
Exercise sheet 3
(Remark (27.10.2024): to solve Question 2.4, use the properties of the trace from the new Section 2.5 of the lecture notes.) |
October 28 | Solutions 3 |
| Exercise sheet 4 | November 11 | Solutions 4 |
|
Exercise sheet 5
(Remark (18.11.2024): corrected the statement of Exercise 5.2.) |
November 25 | Solutions 5 |
|
Exercise sheet 6
(Remark (02.12.2024): corrected the statement of Exercise 4.) |
December 16 | |
| Exercise sheet 7 | No due date (vacation exercises...) |
| Day | Content |
|---|---|
| 18.9.2024 | Quick introduction to number theory. Looking at sums of two squares and asking questions. Fermat's Theorem on primes which are sums of two squares, with the proof of Heath-Brown and Zagier. |
| 20.9.2024 | Some more discussion about sums of two squares. The distribution of primes: statement and proof of Chebychev's estimates, using binomial coefficients and the p-adic valuation of factorials. Discussion of other problems about primes (twin primes, gaps between primes). |
| 25.9.2024 | The twin-prime conjecture; guessing the order of magnitude using heuristic reasoning. Discussion of modular roots of integral polynomial equations in one variable. Statement of Kronecker's Theorem. Link with Fermat's Theorem, the example of cyclotomic polynomials. Quadratic congruences. Statement of the law of quadratic reciprocity. |
| 27.9.2024 | Exercises |
| 2.10.2024 | Comments on Quadratic Reciprocity. Multiplicativity of the Legendre symbol. One and a half proofs of quadratic reciprocity using Gauss sums. |
| 4.10.2024 | Last comments on quadratic reciprocity. First steps in algebraic number theory: definition of number fields and their rings of integers. The additive group structure of the ring of integers. |
| 9.10.2024 | Examples of rings of integers: quadratic fields, cyclotomic fields. The ring of integers is a Dedekind domain: statement and verification of the necessary ring-theoretic properties. Example of failure of unique factorization in ideals for subrings which are not the full ring of integers. Multiplicativity of the norm in the ring of integers. |
| 11.10.2024 | Exercises |
| 16.10.2024 | Important steps in the proof of unique factorization in prime ideals. Example of quadratic fields, recovering Fermat's Theorem on sums of two squares. Definition and finiteness of the class group. Some numerical data for quadratic fields. |
| 18.10.2024 | More discussion of class groups. Factoring ideals generated by prime numbers; split, inert, ramified primes. How to determine the factorization concretely using quotient rings. Characterization of ramified primes with the discriminant. The Kummer-Dedekind theorem. |
| 23.10.2024 | Proof of the Kummer-Dedekind Theorem. Examples. Galois extensions: transitivity of the Galois action of prime ideals dividing a given prime number, and definition of the Frobenius automorphism. |
| 25.10.2024 | Exercises |
| 30.10.2024 | Existence of the Frobenius automorphism. Conjugacy for different prime ideals. The Frobenius for cyclotomic fields. Decomposition in cycle type of the Frobenius. Example of application. |
| 1.11.2024 | Statement of Dirichlet's Unit Theorem. Definition and kernel of the logarithmic embedding of the units. Quadratic reciprocity revisited from the point of view of number fields. A few words on class field theory. |
| 6.11.2024 | Some last comments on algebraic number theory. Introduction to analytic/probabilistic number theory. Arithmetic functions: definition, examples. Counting squarefree numbers. |
| 8.11.2024 | Exercises |
| 13.11.2024 | The average number of divisors: Dirichlet's hyperbola method. Dirichlet convolution, Dirichlet series. Möbius inversion. Examples. |
| 15.11.2024 | Euler products. Averaging arithmetic functions using Dirichlet series: the general approach. Mellin transform, Mellin inversion formula. |
| 20.11.2024 | Averaging arithmetic functions using Dirichlet series: the example of powers of the divisor function. Summation by parts, analytic continuation of the Riemann zeta function. |
| 22.11.2024 | Exercises |
| 27.11.2024 | End of the proof of the average of powers of the divisor function. Discussion of the strategy for the prime number theorem. Primes in arithmetic progressions: statement of Dirichlet's Theorem, quick discussion of uniformity issues. |
| 29.11.2024 | Remarks on Dirichlet's theorem (equidistribution, interpretation as a special case of the Chebotarev Density Theorem). Characters of finite abelian groups. Dirichlet characters. |
| 4.12.2024 | Dirichlet L-functions: definition, analytic continuation, non-vanishing at 1. End of the proof of Dirichlet's Theorem. |
| 6.12.2024 | Remarks and comments on the Prime Number Theorem in arithmetic progressions, especially the importance of uniformity in applications (class number problem, smallest prime in arithmetic progression). The Prime Number Theorem and its variant for the Möbius function using Iwaniec's proof. |
| 11.12.2024 | Lattices, Minkowski's Theorem on the intersection of lattices and symmetric convex sets. Application to the proof of Dirichlet's unit theorem. |
| 13.12.2024 | Exercises |
| 18.12.2024 | Various Questions about number theory. Survey of Thue's Theorem and its application to diophantine equations. |