https://www.youtube.com/impabr
Assistant: Gabriel Fazoli. Register in his list:https://forms.gle/rT1TFj5VYiwviHyd9
Sometimes he will use the link meet.google.com/ewc-ouuo-nyk for the exposition of exercises.Chapter 1 [Ireland-Rosen], Section 1, Unique factorization for integers, page 3,4,5,
Chapter 2, [Ireland-Rosen] Section 1, Theorem 1 (Euclid), there are infinitely many primes in Z. Section 2 Arithmetic functions, page 18,19,20, Moebius inversion theorem. page 21, 22,23,24,25, Section 3, Theorem 3, sum 1/p diverges, Section 4, Growth of pi(x), page 22,23,24,25,
Exercises: [IrelandRosen] 1.23, 1.25, 1.26, 1.30, 2.4, 2.5, 2.6, 2.10, 2.11, 2.15, 2.26, 2.27. [Brochero-Moreira-Saldanha-Tengan] 1.1, 1.3, 1.5, 1.12, 1.19, 1.20, 1.21.
Chapter 3 [Ireland-Rosen, pages 28-40] Congruence of integers mod n, units and linear equations (prop 3.3.1,3.3.2), Euler and Fermat's Little Theorem, systems of linear equations (Chinese Remainder Theorem, section 4), Wilson's theorem (prop 4.1.1 and Corollary).
Chapter 4 [Ireland-Rosen, pages 40-45] Roots of unity mod p (prop 4.1.2), primitive roots mod p (theorem 1), primitive roots mod a power of p (theorem 2, 2'), structure of units mod n (theorem 3), classification of numbers admitting primitive roots (prop 4.1.3).
Exercises: [IrelandRosen]. 3.1, 3.16, 3.21, 3.23, 3.24, 3.25, 3.26, 4.7, 4.15, 4.24. [Brochero-Moreira-Saldanha-Tengan] 1.33, 1.34, 1.36, 1.46, 1.49, 1.60, 1.77, 1.79, 1.80, 1.82
Chapter 5 [Ireland-Rosen, pages 50-60] Quadratic residues (prop 5.1.1), Legendre symbol and computations (prop 5.1.2, Corollary 3, Gauss' Lemma, prop 5.1.3), Quadratic Reciprocity Law (theorem 1 and proof, section 3).
Chapter 1 [Ireland-Rosen, pages 8-13] The ring of Gauss' integers (prop 1.4.1), Chapter 4 [Garcia-Lequain, pages 101-108] Primes that can be written as sum of two squares (theorem IV.1.2), irreducible elements of Z[i] (Corollary IV.1.3), Fermat's Theorem on numbers that can be written as sum of two squares (theorem IV.1.6), Pythagorean triples (theorem IV.2.2).
Chapter 4 [Brochero-Moreira-Saldanha-Tengan, pages 126-139] Legendre's Theorem (theorem 4.4), Gauss' Three Squares Theorem (theorem 4.15).
Exercises: [IrelandRosen] 5.3, 5.7, 5.8, 5.11, 5.13, 5.15, 5.16, 5.18, 5.19. [Brochero-Moreira-Saldanha-Tengan] 1.51, 1.52, 1.53, 1.81, 1.83, 2.7, 2.8, 2.12, 2.19, 2.20, 2.22.
Chapter 5 [Martin, pages 31-35] The ring Z[\sqrt{n}] (Lemma 5.1, 5.2, Bramhagupta's composition rule), units of Z[\sqrt{n}] (Lemma 5.5, theorem 5.6), Dirichlet's aproximation Theorem (prop 5.8), solution to Pell's equation (theorem 5.9), Chapter 4 [Brochero-Moreira-Saldanha-Tengan, pages 140-142] Minkowski's Theorem (theorem 4.18), Sums of four squares (theorem 4.20).
Chapter 17 [Ireland-Rosen, pages 284-288] The ring Z[w] (prop 1.4.2), Fermat's equation for exponent 3 (prop 17.8.1), cubic equation with infinite rational points (prop 17.9.1).
Chapter 12 [Ireland-Rosen, pages 174-180] Unique factorization rings, Rings of integers of a number field, Dedekind rings and unique factorization of ideals, the class group.
Chapter 14 [Ireland-Rosen, pages 215-216] Some properties of cyclotomic fields (section 5, Lemma 1, 2), Chapter 1 [Washington, pages 1-7] The First case of Fermat's Last Theorem for regular primes.
Exercises, see the link
Week 7, [Hossein Movasati], Basics
of complex analysis and holomorphic functions, absolute convergence,
Chapter 16 [Ireland-Rosen], Riemann's zeta function, Dirichlet
character, Dirichlet L-functions, see also the chapter on Zeta
function of my
lecture notes
Exercises: Chapter 16,
[Ireland-Rosen] Ex. 2,3,4,5,6,8,9,10,13,14,15.
Problem 1, Configuration of lines
Problem 2, Differential equations and number theory
Problem 3, A big matrix: How to compute its rank? I
Problem 4. A big matrix:How to compute its rank? II