Abstract: We will introduce algebraic structures as sets with operations of various arities subject to certain axioms. We study groups as the first example. We prove some standard results as the Lagrange theorem or the simplicity of the group A5. We investigate cyclic groups and we apply our results to find simple proofs of some standard number theoretic theorems. We study operations of groups on sets and we introduce Polya's theory of counting. Then we pass to commutative rings and divisibility. We define and study PIDs and UFDs. We will apply the results to solve some non-trivial Diophantine equations. We will conclude the course with the introduction to the theory of fields.
Abstrakt: Program proseminare bude behem semestru upresnen a bude zaviset na zajmu studentu. Zacneme klasifikaci konecne generovanych Abelovych grup a nasledne konecne generovanych modulu nad obory hlavnich idealu.
Abstrakt: V prvnim semestru bude prednaska probihat formou kontrolovane cetby. Student se seznami s obsahem textu [4] na konci teto stranky.
Abstract: Po prvnim seznameni se svazy, distributivitou a modularitou se podivame na algebraicke svazy a jejich vztah k uzaverovym systemum. Po te definujeme volne svazy a budeme v nich resit problem slov. Znovu, ale ted podrobneji prozkoumame distributivni a modularni svazy. Popiseme kongruence svazu, a ukazeme jak kongruencni distributivitu svazu aplikovat pri studiu svazovych variet.
