Module page
Algebra I
| academic year: | 2013/2014 |
| instructor: | Ernesto Spinelli |
| degree course: | Mathematics - DM 270/04 (triennale), I year |
| type of training activity: | di base |
| credits: | 9 (72 class hours) |
| scientific sector: | MAT/02 Algebra |
| teaching language: | italiano |
| program: | I-Z |
| period: | II sem (03/03/2014 - 13/06/2014) |
Lecture meeting time and location
Presence: highly recommended
Module subject:
- The language of sets
Elements of set theory. Maps. Equivalence and order relations. Quotient set and canonical projection. Cardinality of sets. - Arithmetic on Z and modular arithmetic.
Euclidean division in Z. Greatest common divisor. The Euclidean algorithm for the computation of the GCD. The fundamental Theorem of arithmetic. Congruences. Units of Z_m. Eulero-Fermat Theorem. Fermat's little Theorem. Wilson Theorem. The chinese remainder Theorem. - Groups
Algebraic structures: definitions and examples. Subgroups and normal subgroups of a group. Group homomorphisms. Quotient group. Homomorphism and correspondence theorems. Lagrange Theorem and Cayley Theorem. C_n, S_n, D_n, linear groups. Permutations and conjugation. - Rings and factorization
Examples: integral domains and fields. Ideals of a ring. Ring homomorphisms. Quotient ring. Homomorphism and correspondence theorems. Rings of polynomials and their universal property. Polynomials with coefficients in a domain. Field of fractions of an integral domain. Euclidean domains: examples. The ring of Gauss integers. Principal ideal domains. Prime and maximal ideals. Irreducible polynomials. Prime and irreducible elements of a domain. Unique factorization domains. Gauss Lemma. Eisenstein Criterion. The irreducible elements of Z[x]. Unique factorization in Z[x]. Primes in the ring of the Gauss integers.
Detailed module subject: Diario delle lezioni completo
Suggested reading:
M. Artin, "Algebra", Boringhieri
I.N. Herstein, "Algebra", Editori Riuniti.
G.M. Piacentini Cattaneo, "Algebra, un approccio algoritmico", Decibel/Zanichelli.
G. Campanella, "Appunti di Algebra 1" ed esercizi.
Type of course: standard
Exercises:
- Foglio 1 (Esercitazione 7 marzo)
- Esercizi proposti (A.A. 2012/13): http://www.mat.uniroma1.it/people/carlucci/1213/algebra
- Esercizi proposti (A.A: 2011/12)
- Foglio 2 (Esercitazione 14 marzo)
- Foglio 3 (esercitazione 21 marzo)
- Foglio 4 (Esercitazione 28 marzo)
- Foglio 5 (Esercitazione 4 aprile)
- Foglio 6 (Esercitazione 11 aprile)
- Foglio 7 (Esercitazione 9 maggio)
- Foglio 8 (Esercitazione 16 maggio)
- Foglio 9 (Esercitazione 23 maggio)
- Foglio 10 (Esercitazione 30 maggio)
Examination tests:
Knowledge and understanding: Understand the meaning of abstract structure and identification up to isomorphism. Use of equivalence to define new objects and new structures. Recognize the cardinality of a set. Understand the difference between prime and irreducible element in a domain and the importance of the factorization Theorem. Use the first notions about groups and order of elements.
Skills and attributes: Verify the properties of a map. Verify the properties of a relation. Compute the GCD in Z, Z[i] and K[x] through the Euclidean algorithm. Solve linear congruences and apply Chinese remainders theorem. Study the factorization of polynomials in K[x] in simple cases. Describe the structure of some classes of groups (cyclic, symmetric and dihedral groups). Apply the Homomorphism Theorem.
Personal study: the percentage of personal study required by this course is the 65% of the total.