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Ring Theory and Applications

Code: M2012     Acronym: M2012     Level: 200

Classification Keyword
OFICIAL Mathematics

Instance: 2019/2020 - 2S Ícone do Moodle

Active? Yes
Responsible unit: Department of Mathematics
Course/CS Responsible: First Degree in Mathematics

Cycles of Study/Courses

Acronym No. of Students Study Plan Curricular Years Credits UCN Credits ECTS Contact hours Total Time
L:B 0 study plan from 2016/17 3 - 6 56 162
L:CC 1 Plano de estudos a partir de 2014 2 - 6 56 162
L:F 0 study plan from 2017/18 2 - 6 56 162
L:G 1 study plan from 2017/18 2 - 6 56 162
L:M 9 Plano estudos a partir do ano letivo 2016/17 2 - 6 56 162
L:Q 0 study plan from 2016/17 3 - 6 56 162

Teaching Staff - Responsibilities

Teacher Responsibility
Carlos Miguel de Menezes

Teaching - Hours

Theoretical classes: 0,00
Theoretical and practical : 0,00
Type Teacher Classes Hour
Theoretical classes Totals 1 0,00
Carlos Miguel de Menezes 2,00
Theoretical and practical Totals 1 0,00
Carlos Miguel de Menezes 2,00

Teaching language



Familiarization with basic ring, field, and Galois theory and some of its applications.

Learning outcomes and competences

Understand the basic concepts and results of ring and field theory, including homomorphisms, ideals, polynomial rings, field extensions, the structure of finite fields, and their applications, namely in cryptography and error correcting codes.

Working method


Pre-requirements (prior knowledge) and co-requirements (common knowledge)

It is recommended that students previously take courses introducing abstract algebraic concepts such as "Teoria de Grupos" and "Álgebra Linear e Geometria Analítica I".


Fundamental concepts and  threorems of Ring theory,  Field extensions, and Galois theoy:

  Important examples of unital rings and fields. Among them
 Z (integer numbers), Q (rational number) , R (real numbers), C (complex numbers), H (quaternions), matrix rings over a unital ring, rings of quaratic integers, fields of quadratic integers, polynomial rings, formal power series rings. Groups of units of a unital ring.
Some remarquable elements of a ring: units, zero divisors, nilpotents,  idempotens, unipotents)    Subrings,, lateral and bilateral ideals.

2. Homomorphisms of unital rings. Study of  automorphism group of some rings.

3. Subrings, lateral and  bilateral ideals.

4. Main constructions of rings: subrings,  quotient by an ideal, direct product, , subring generated by a subset. Noether isomoprhisms and correspondence theorem.

5. Divisibility in ring. Prime and irreductible elements of a ring. Prime and maximal  ideals. Integral domains, principal ideal domains, euclidean doimains, Bézout domains, Unique factorization domains.  Criteri for irredutibility of polynomials. Symmetric Polynomials.

6. Field extensions. Structure of finite fields. 
7. Soluble Groups. Galois theory

8. If time allows some appications to cryptography and error correcting codes will be illustrated.


Mandatory literature

Carlos Menezes; Apontamentos de Teoria de Anéis e Aplicações-2019-2020

Complementary Bibliography

David S. Dummit; Abstract algebra. ISBN: 0-13-005562-X

Teaching methods and learning activities

Presentation of the course material by the teacher. Solution and discussion of exercises.


Physical sciences > Mathematics > Algebra

Evaluation Type

Evaluation with final exam

Assessment Components

designation Weight (%)
Exame 100,00
Total: 100,00

Amount of time allocated to each course unit

designation Time (hours)
Estudo autónomo 106,00
Frequência das aulas 56,00
Total: 162,00

Eligibility for exams

No restrictions.

Calculation formula of final grade

Final exam only.

Special assessment (TE, DA, ...)

Any type or special examination can be from one the following types: exclusively by an oral examination,  only  a written exam, one oral examination and a written exam.

The decision of which of the above types is each special examination  is exclusively the responsability of the teacher  assigned to the curricular unit.

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