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If the field of fractions Q is replaced with its finite extension K, for example K=Q(i) or K=Q(2^{1/2}), the ring of integers Z extends in the ring of integers O_K of K. Algebraic number theory investigates arithmetic of O_K, especially unique factorization. These results have important applications in the original ring Z, mainly for solving diophantine equations. In the course we shall present basic notions and results as well as some applications to diophantine equations.
Last update: T_KAM (27.04.2005)
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The topic is algebraic techniques in combinatorics and number theory. We will learn many interesting results obtained by them. Last update: Klazar Martin, doc. RNDr., Dr. (21.09.2016)
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Oral exam, in person or in distant mode. Last update: Klazar Martin, doc. RNDr., Dr. (22.09.2020)
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Borevich and Shafarevich: Number Theory, Academic Press 1966. This is one of the references but I draw from many others, such as the survey article of N. Alon on algebraic methods in combinatorics. Further literature depends on the current lecture and will be given in the lecture. Last update: Klazar Martin, doc. RNDr., Dr. (12.10.2017)
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Exam is oral, with written preparation. For concrete requirements see the above homepage of the course. An example of exam questions: 1. Prove that there are infinitely many primes of the form p = 1 + mn. 2. Prove Wedderburn's theorem on skew fields. 3. Prove Fermat's last theorem ... for polynomials. 4. Prove the theorem of Ko Chao (if q > 3 is a prime number then x^2 - y^q = 1 has no solution in positive integers x, y). 5. Prove the Chevalley-Warning theorem and the corollary on multigraphs. 6. Prove Alon's Combinatorial Nullstellensatz and the corollary on hyperplanes. Last update: Klazar Martin, doc. RNDr., Dr. (11.06.2019)
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Basic techniques and results from algebra applied in combinatorics and number theory. For example, in extremal problems (combinatorics) or diophantine equations (number theory). Last update: Klazar Martin, doc. RNDr., Dr. (21.09.2016)
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