Unit COMMUTATIVE AND COMPUTATIONAL ALGEBRA
- Course
- Mathematics
- Study-unit Code
- 55A00034
- Curriculum
- Matematica per la crittografia
- Teacher
- Giuliana Fatabbi
- Teachers
-
- Giuliana Fatabbi
- Hours
- 63 ore - Giuliana Fatabbi
- CFU
- 9
- Course Regulation
- Coorte 2022
- Offered
- 2022/23
- Learning activities
- Caratterizzante
- Area
- Formazione teorica avanzata
- Academic discipline
- MAT/02
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Theory of commutative unitary rings and introduction to the theory of Groebner basis
- Reference texts
- Atiyah-Macdonald, Intoduction to Commutative algebra, Addison-Wesley, 1969
Cox-Little-O'Shea, Ideals, Varieties, and Algorithms, Springer , 1997
Any supplementary material available in Unistudium - Educational objectives
- The aim of the course is to deepen the theory of commutative rings with units, with particular attention to the ring of polynomials and its quotients, having in view the applications of commutative algebra to algebraic geometry and to introduce the theory of the bases of Groebner, in order to introduce the student to computational algebra and its applications.
The course also aims to refine the abstraction skills and, on the other hand, show how a good theoretical knowledge allows the development of significant application tools.
In particular, the course aims to make students acquire the following skills:
-Knowledge and understanding:
-Knowledge of results and fundamental methods of the theory of algebraic structures and its applications.
-Ability to read, understand and deepen a topic of mathematical literature and propose it again clearly and accurately.
-Ability to understand problems and to extract their substantial elements.
Applying knowledge and understanding:
-ability to build or solve examples or exercises, to solve mathematical problems which, although not common, are of a similar nature to others already known and to face new theoretical problems, researching the techniques and applying them appropriately.
-Develop mathematical skills in reasoning, manipulation and calculation; have adequate computational skills, including knowledge of specific software.
Making judgments:
-being able to construct and develop logical arguments with a clear identification of assumptions and conclusions;
-be able to recognize correct proofs, and to identify fallacious reasoning.
-be able to produce proposals capable of correctly interpreting complex problems in the field of algebra and its applications.
-being able to autonomously formulate pertinent judgments on the applicability of algebraic models to theoretical and / or concrete situations.
Communication skills:
-Ability to present arguments, problems, ideas and solutions, both one's own and others, in mathematical terms and their conclusions, with clarity and accuracy and in ways appropriate to the listeners to whom one is addressing, both in form oral and in written form.
-Ability to clearly motivate the choice of strategies, methods and contents, as well as the computational tools adopted.
Learning skills:
Read and deepen a topic of algebraic literature. To deal autonomously with the systematic study of algebraic topics not previously explored. - Prerequisites
- Basic concepts of rings and ideals, in particular ring of polynomials in one indeterminate over a field.
- Teaching methods
- Face to face lessons
- Other information
- According with the students attending the course, the course can be partially or entirely given in English. The exam can be in English, upon request of the student.
Usage of platform "uni-studium" - Learning verification modality
- There will be an oral exam lasting 45-60 minutes. During this test the student will be asked to illustrate the solution of 2 or 3 exercises assigned at the end of the course and to explain some topics covered in the course. This oral test tends to assess the level of understanding of the topics covered and of critical study and personal re-elaboration.
Upon request, the exam can be taken in English.
For information on support services for students with disabilities and / or SLD, visit the page http://www.unipg.it/disabilita-e-dsa. The teacher is in each
case available to personally evaluate, in specific cases, any compensatory measures and / or personalized paths in the case of students with disabilities and / or SLD. The teacher is also available to evaluate any personalized courses for working or non-attending students - Extended program
- I. Rings and ideals. First properties of commutative rings with unity. Prime ideals and maximal ideals. Local rings. Nilradical and Jacobson's radical. Operations with ideals; radical of an ideal. Homomorphisms. Extended ideals and contracted ideals.
II. Forms. Definition and first properties. Direct product and direct sum: free modules. Finitely generated modules and Nakayama's lemma. Homomorphisms between modules. Algebras.
III. Rings and modules of fractions. Definition and properties. Localization and local ownership. Ideal in rings of fractions.
IV. Noetherian rings. Affine varieties, affine K-algebras and basic dictionary of algebra-algebraic geometry. Krull size. Noetherian rings and modules: definitions and first properties.
V. Artinian rings. Artinian rings and modules. A ring is Artinian if and only if it is Noetherian and has zero dimension.
YOU. Primary decomposition. Primary ideals; primary decomposition. First associates and their characterization. Divisors of zero. Uniqueness of the isolated components. The Noetherian case.
VII. Hilbert's zero theorem: weak form and strong form.
VIII. Integral dependence. Definitions and first properties. Going Up Theorem. Normal domains and the Going Down Theorem.
IX. Outline of dimension theory. Chains of prime, height, dimension. Krull's principal ideal theorem. Krull's height theorem. Size of rings of coefficient polynomials in a field. Local rings. Parameter system. Immersion size. Regular local rings (only definition and geometric importance).
X. I. Basic theory of Groebner Bases. The linear case. The one-variable case. Monomial orderings. The division algorithm. Basic definition of Groebner. S-polynomials and Buchberger's algorithm. Reduced Groebner bases.
XI. Applications of Groebner Bases. Elementary applications of Groebner Bases. Elimination theory. Polynomial maps. Some applications to Algebraic Geometry