| Code |
15626
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| Year |
1
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| Semester |
S1
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| ECTS Credits |
6
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| Workload |
TP(45H)
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| Scientific area |
Mathematics
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Entry requirements |
Does not have.
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Learning outcomes |
Upon completion of this curricular unit, the student will
1. have a good understanding of the key concepts, results and techniques of Galois Theory and Commutative Algebra that occur in other areas of Mathematics and its Applications, namely, in Algebraic Geometry, Number Theory, Coding Theory and Cryptography;
2. be able to use mathematical language correctly in oral and written communication, presenting the studied topics,solved exercises and proposed assignments with clarity, precision and confidence.
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Syllabus |
1. Galois Theory
1.1. Galois group
1.2. Normal extensions and separable extensions
1.3. Galois correspondence
1.4. Solvability of algebraic equations by radicals
2. Commutative Rings and Ideals
2.1. Definitions and elementary properties
2.2. Prime ideals and maximal ideals
2.3. The nil radical and the Jacobson radical
2.4. Noetherian rings and Artinian rings
2.5. Structure theorem for Artinian rings
2.6. Hilbert's zeros theorem
3. Modules
3.1. Modules and module homomorphisms
3.2. Exact sequences
3.3. Submodules and quotient modules
3.4. Direct sums and direct products
3.5. Linear independence
3.6. Tensor products
3.7. Modules over integral domains
3.8. Finitely generated modules
3.9. Applications: diagonalization of matrices over principal ideal domains; classification of finite abelian groups;Jordan canonical form
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Main Bibliography |
Stewart, I. (2015). Galois theory. Fourth edition. CRC Press, Boca Raton, FL.
Artin, E. (1998). Galois Theory, Dover Publications, Mineola, NY.
Tignol, J. P. (2016). Galois theory of algebraic equations. Second edition. World Scientific Publishing Co Pte Ltd.
Atiyah, M. F., MacDonald, I. G. (1969). Introduction to Commutative Algebra. Addison-Wesley Publishing Co., Reading,Mass.-London-Don Mills.
Zariski, O., Samuel, P. (1975). Commutative Algebra. Springer Science & Business Media.
Eisenbud, David (1995). Commutative algebra with a view toward algebraic geometry. Graduate Texts in Mathematics,150. Springer-Verlag, New York.
Kaplansky, I. (1970). Commutative Rings. Boston.
Kemper, G. (2010). A Course in Commutative Algebra. Springer Science & Business Media.
Matsumura, H. (1980).Commutative Algebra. Second edition. Mathematics Lecture Note Series, 56.Benjamin/Cummings Publishing Co., Inc., Reading, Mass.
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Teaching Methodologies and Assessment Criteria |
1. The classes, which are a combination of theory and practice, are based on the main bibliographic references pertaining to Galois Theory and Commutative Algebra;
2. Further reading on some topics of the syllabus is suggested;
3. Problem solving and group discussions about methods and techniques used are encouraged;
4. An assignment on a topic of the syllabus and respective presentation is proposed.
Assessment may be carried out during the teaching-learning process and consists of an assignment worth 20% of the total grade and a final test worth 80% of the total grade. Additionally, the student may be assessed by a final exam which comprises the total grade.
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Language |
Portuguese. Tutorial support is available in English.
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