Algebra 3

A.Y. 2020/2021
Overall hours
Learning objectives
The course concerns fields theory and Galois theory, with an introduction to Algebraic Number Theory.
Expected learning outcomes
Knowledge of the main results of Galois Theory and some basic notion of Algebraic Number Theory. Ability of computing the lattice of the subfields a field extension, the Galois group of a Galois extension and ability of applying the basic notions of Algebraic Number Theory.
Course syllabus and organization

Single session

Lesson period
First semester
The lessons will be delivered online by means of the virtual Zoom rooms furnished by the department. They will be recorded and they will be available on Ariel. Other communications will take place through Ariel.
Course syllabus
The concept of trascendence and algebraicity over a field. Existence of algebric closures. Separable and Galois extensions. Fundamental theorem of Galois theory. Finite fields. Cyclotomic ancd cyclic extensions. Galois solvability theorem. Rudiments about the inverse problem (Hilbert irreducibility theorem) and Algebraic Number Theory.
Prerequisites for admission
Basic knowledge of algebra (Algebra 1-2).
Teaching methods
See didattica in fase emergenziale.
Teaching Resources
-S. Lang, "Algebra", Springer-Verlag, 2002.
-J. S. Milne, "Fields and Galois Theory", Version 4.61, April 2020. Available at
-F. Andreatta e M. Bertolini, "Appunti di Teoria dei Numeri". Available at
Assessment methods and Criteria
The final examination consists of a written exam and an oral discussion, to be given in the same session. It is not allowed to use notes, books or calculators, except differently stated. The students that passed positively the midterm exam have the right to skip the part of the exam concerning the first part of the course in the written exams of January and February, not the next sessions. In this case, they will have less time with resplect to those that have to do the total exam and the grade of the written exam will be proportional.
Important: the above modality of examination may be subject to changes due to the sanitary emergency. The changes will be communicated during the lessons and the information will be reported on Ariel.
MAT/02 - ALGEBRA - University credits: 9
Practicals: 48 hours
Lessons: 45 hours
Educational website(s)
On appointment
Via Cesare Saldini 50