# Algebra and Number Theory

## General data

Course ID: | 0600-MS2-1ATL | Erasmus code / ISCED: | 11.103 / (unknown) |

Course title: | Algebra and Number Theory | Name in Polish: | Algebra i teoria liczb |

Department: | (in Polish) Zakład Algebry | ||

Course groups: |
(in Polish) 2L stac. II st. studia matematyki - przedmioty obowiązkowe |
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ECTS credit allocation (and other scores): |
(not available)
view allocation of credits |
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Language: | Polish | ||

Type of course: | obligatory courses |
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Requirements: | Algebra I 0600-MS1-2ALG1 |
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Short description: |
Course objectives: The skill to use the notation of an algebraic field extension, to find the Galois group of field extensions and intermediate fields of Galois extensions, the expertise of a complete characterization of Galois extension and fundamental theorems of the Galois Theory, the skill to notice the Galois Correspondence between a field extension and its Galois group, the understanding of the importance of the Galois Theory in the solution of the problem of solving polynomial equations by radicals, and of the Three Geometric Problems of Antiquity, the skill to solve polynomial equations of both 3rd and 4th degree, the expertise of the Cardano Formula; the skill to use the notion of an algebraic integer, to solve Diophantine equations using the unique factorization of the ring of algebraic integers in some quadratic number fields, to find an integral basis of both quadratic number fields and cyclotomic fields; the skill to discuss the problem of the distribution of prime numbers |
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Full description: |
Course profile: academic Form of study: stationary Course type: obligatory Academic discipline: Mathematics, field of study in the arts and science: mathematics Year: 1, semester: 1 Prerequisities: none lecture 30 h. exercise class 30 h. Verification methods: lectures, exercises, consultations, studying literature, home works, discussions in groups. ECTS credits: 5 Balance of student workload: attending lectures15x2h = 30h attending exercise classes 15x2h = 30h preparation for classes 7x3h = 21h completing notes after exercises and lectures 7x2h = 14h consultations 12x1h = 12h the final examination: preparation.and take 12h + 3h = 15h control works: repeating the material and preparation 3x4h = 12h Quantitative description Direct interaction with the teacher: 75 h., 3 ECTS Practical exercises: 77 h., 3 ECTS |
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Bibliography: |
(in Polish) 1. Andrew Baker, An introduction to Galois Theory (writen for a University of Glasgow level 4 Honours degree course). http://www.maths.gla.ac.uk/~ajb/dvi-ps/Galois.pdf 2. Jerzy Browkin, Teoria ciał, Wydawnictwo Naukowe PWN, Warszawa 1977. 3. Maciej Bryński, Elementy Teorii Galois, Wydawnictwo Alfa, Warszawa 1985. 4. Ian Steward, Galois Theory, third edition, Chapman & Hall/CRC, A CRC Company Boca Ration, London, New York 2004. |
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Learning outcomes: |
Learning outcomes: A student uses the notion of an algebraic field extension; knows a complete characterization of a Galois extension and fundamental theorems of the Galois Theory; finds the Galois group of field extensions and intermediate fields of Galois extensions; understands the importance of the Galois Theory in the solution of the problem of solving polynomial equations by radicals, and of the Three Geometric Problems of Antiquity.K_W04, K_W05, K_W07, K_W08, K_U01, K_U02, K_U03, K_U04, K_U10, K_U13, K_U14 A student examines the unique factorization of the ring of algebraic integers in quadratic number fields; solves Diophantine equations using the unique factorization of the ring of algebraic integers in some quadratic number fields; understands the importance of the unique factorization of the ring of algebraic integers in some quadratic number fields in the solution of some problems in the Number Theory.K_W04, K_U01, K_U02, K_U03, K_U04, K_U10, K_U13, K_U14 A student discusses the problem of the prime numbers distribution.K_W04, K_U01, K_U02, K_U03, K_U04, K_U13, K_U14 A student obtains basic skills of creative development of algebra.K_K01, K_K02, K_K07 A graduate understands that modern technologies result from scientific discoveries including algebra.K_K08, K_K05 |
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Assessment methods and assessment criteria: |
The overall form of credit for the course: final exam |

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