|Responsible unit:||Department of Mathematics|
|Course/CS Responsible:||First Degree in Mathematics|
|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|
|Carlos Miguel de Menezes|
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.
|Frequência das aulas||56,00|
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.