[01/04/19] On April 4, 2019 the lecture will not take place
[26/02/19] Il corso inizierà il 26 Febbraio 2019
Diario delle Lezioni: (following Schoof's Notes)
[26-2-2019] Chapter One Introduction: The Pythagorean equation, P. de Fermat theorem, Fermat’s last theorem, prove the only solution $x,y\in\mathbb{Z}$ of the equation $x^{3}=y^{2}+1$ is given by $x=1,y=0$.
[27-2-2019] Chapter two Number fields: Theorem of the primitive element, relation between the degree of the number field $F$ also basis of $F$ with distinct field,
homomorphisms $\phi:F\rightarrow\mathbb{C}$, examples.
[28-2-2019] Chapter two+three Maps a $\mathbb{Q}$-basis of the number field $F$ to an $\mathbb{R}$-basis of $F\otimes\mathbb{R}$, Cyclotomic fields, Norms, Traces and The characteristic polynomial, examples.
[05-3-2019] Chapter three
The Discriminant of the basis of the number field $F$, the Resultant of two polynomials, relation between discriminant, norm, characteristic polynomial and Resultant.
[06-3-2019] Chapter Four
Rings of integers: Roll of the Integral element in the minimal and characteristic polynomial over $\mathbb{Q}$ in $\mathbb{Z}[T]$ and finitely generated additive subgroup of $F$, prove the set $\mathfrak{O}_{f}$ is subgroup of $F$, the ring integer of quadratic filed.
[07-3-2019] Chapter Four
The discriminant of the basis of $\mathfrak{O}_{f}$, the Ideal $I$ of $\mathfrak{O}_{f}$, examples.
[13-3-2019] Chapter Five
Dedekind rings: Noetherian ring, characterize of Noetherian ring, Kurll Dimension, Fractional ideal. (By: Florian Luca)
[14-3-2019] Chapter Five proof In Dedekind ring The set $Id(R)$ is an abelian group and every fractional ideal can
be written as a finite product of prime ideals.(By: Florian Luca)
[19-32019] Appendix of Chapter Four Computing the ring of integer and the Discriminant. (By: Fouotsa Tako BORIS)
[20-3-2019] Chapter Six
&The Dedekind $\zeta$-function: $\mathfrak{O}_{F}$is a Dedekind ring, proof of the norm of two non zero ideals of $\mathfrak{O}_{F}$ is multiplicative, prime decomposition of the ideal generated by $p$ in $\mathfrak{O}_{f}$ and classification to totally split, totally ramified and remains prime(inert).
[27-3-2019] Chapter Six The factor some small prime numbers into prime ideals in $\mathbb{Z}[\sqrt{-5}]$, The Dedekind $\zeta$-function.
[28-3-2019] Chapter Seven Finitely generated abelian groups: The structure of finitely generated abelian groups, the relation between induces of finitely generated free groups and determinants.
[02-4-2019] Discussion of problems from Chapter 6 & 7
[16-4-2019] Chapter Nine Discriminant and ramification: Kummer's decomposion lemma, Chebotarev density Theorem and examples
[17-4-2019] Chapter Nine Proof then if $p$ is ramified then $p$ divides the discriminant, Dedekind's Criterion and examples.
[18-4-2019] Chapter Nine
Proof that the ring of integer of $\mathbb{Q}(\zeta_{p^{n}})$ is $\mathbb{Z}(\zeta_{p^{n}})$, Dedekind Theorem.
[02-5-2019] Chapter Ten Minkowski Convex Body Theorem on the Geometry of Numbers and its application to Number Fields on the determination on integers in ideals $I$ with norm bounded in terms of $N(I)$ and the other invariants of the field.
[07-5-2019] Chapter Ten Lowerbounds for the discriminants of a number field $F$ in terms of the the degree and the number of complex embeddings of $F$. Finiteness of the Class Group. Examples $\mathbb{Q}(\alpha), \alpha^3=\alpha+1$ and $\mathbb{Q}(\sqrt{-47})$.
[08-5-2019] Chapter Ten Hermite Theorem on the finiteness of number field with a given discriminant.
[14-5-2019] Chapter Ten & Chapter Eleven End of the proof of Hermite Theorem. Dirichlet Theorem on the structure of units. Beginning of the proof
[16-5-2019] Chapter Eleven Dirichlet Theorem on the structure of units. Continuation of the proof.