Algebra II


Theory of factorization in rings, rings of polynomials and field extensions. 

General characterization





Responsible teacher

Herberto de Jesus da Silva


Weekly - 5

Total - 70

Teaching language



Knowledge corresponding to the contents of Algebra I (1st semester-2nd year).


1. J. Durbin, Modern Algebra, John Wiley & Sons, Inc.

2. N. Jacobson, Basic Algebra I, W. H. Freeman and Company.

3. S. Lang, Algebra, Addison-Wesley Publishing Company, Inc.

4. A. J. Monteiro e I. T. Matos,  Álgebra, um primeiro curso, Escolar Editora.

5. M. Sobral,  Álgebra, Universidade Aberta.

6. G.M.S. Gomes, Anéis e Corpos, uma introdução, DM-FCUL, 2011.

Teaching method

Lectures + problem-solving sessions (5h00). 

Evaluation method



Students enrolled for the first time in the unit must attend all classes, except up to 3 lectures and up to 3 problem-solving classes.

Students that have already been enrolled in the unit must attend, at least, 2/3 of the lectures and 2/3 of the problem-solving classes.

The students that do not fulfill the above requirements automatically fail "Álgebra II". 

There are two mid-term tests. These tests can substitute the final exam if CT is, at least, 9.5.  CT is the arithmetic mean of the non-rounded grades of the tests.

If the student satisfies the conditions above with CT (rounded to units) greater than 16, he may choose between having 16 as final grade or undertake a complementary assessment.

To be approved in final exam, the student must have a minimum grade of 9.5 in it.  Again, for grades (rounded to units) greater than 16, the student must undertake a complementary assessment, otherwise his final grade will be 16.

In tests and in exam, any kind of consultation is not allowed.

In tests and in exam, students are allowed to use sheets of paper and a ballpoint and nothing else.

More detailed rules are available in the portuguese version. 

The non-portuguese students should address the professor to ask any question that is not in this english version.











Subject matter

I. Theory of Factorization

1. Divisors.
2. Prime and coprime elements.
3. Gauss semigroups.
4. Gauss rings.
5. Principal ideal rings.
6. Euclidean domains.

II. Rings of Polynomials

1. Rings of polynomials.
2. Division algorithm.
3. Polynomial functions.
4. Theory of factorization in rings of polynomials.
5. Irreducibility.

III. Field extensions

1. Prime fields. 
2. Extensions. Simple extensions.  Algebraic extensions.
5. Algebraically algebraic closed fields and algebraic closure of a field.
6. Rupture and splitting fields.
7. Finite fields.


Programs where the course is taught: