Algebra III - rings and fields
Summary
This is an introductory course in ring and field theory.
Content
Fundamental concepts, constructions, and theorems:
- Rings, subrings, homomorphisms of rings
- Examples of rings
- Integral rings, fields of fractions
- Ideals, quotient rings and their universal properties, the characteristic of a ring, operations on ideals, correspondence theorems, product of rings, the Chinese remainder theorem
- Prime and maximal ideals
Arithmetics in rings
- Euclidean rings
- Principal rings
- Associated, prime, and irreducible elements
- Factorial rings
- Noetherian rings
- Characterization of factoriality
- Gauss's Lemma and theorem
- Irreducibility criteria
Field Theory:
- Algebras over a field
- Field extensions, algebraic and transcendental elements, the degree of a field extension, algebraic extensions, construction of simple field extensions, splitting fields
- finite fields
- separable extensions, primitive element theorem
- Galois theory
- purely inseparable extensions, separable-inseparable decomposition
- algebraically closed fields, separable closure, inseparable closure
Learning Prerequisites
Required courses
- Algebraic structures
- Linear algebra I and II
- Group theory
Assessment methods
Written exam
In the programs
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Algebra III - rings and fields
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: mandatory
Reference week
Mo | Tu | We | Th | Fr | |
8-9 | |||||
9-10 | |||||
10-11 | |||||
11-12 | |||||
12-13 | |||||
13-14 | |||||
14-15 | |||||
15-16 | |||||
16-17 | |||||
17-18 | |||||
18-19 | |||||
19-20 | |||||
20-21 | |||||
21-22 |
Légendes:
Lecture
Exercise, TP
Project, Lab, other