Algebra and Number Theory

General data

Course ID:	0600-MS2-1ATL
Erasmus code / ISCED:	11.103 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (unknown)
Course title:	Algebra and Number Theory
Name in Polish:	Algebra i teoria liczb
Organizational unit:	(in Polish) Instytut Matematyki.
Course groups:
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	Polish
Type of course:	obligatory courses
Requirements:	Algebra I 0600-MS1-2ALG1 Elementary Number Theory 0600-MS1-1ETL Linear Algebra I 0600-MS1-1AL1
Prerequisites:	Algebra II 0600-MS1-2ALG2 Linear Algebra II 0600-MS1-1AL2
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
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
Bibliography:	(in Polish) 1. Jerzy Browkin, Teoria ciał, Wydawnictwo Naukowe PWN, Warszawa 1977. 2. Maciej Bryński, Elementy Teorii Galois, Wydawnictwo Alfa, Warszawa 1985. 3. Ian Steward, Galois Theory, third edition, Chapman & Hall/CRC, A CRC Company Boca Ration, London, New York 2004.
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
Assessment methods and assessment criteria:	The overall form of credit for the course: final exam

This course is not currently offered.

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