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National Repository of Grey Literature 3 records found  Search took 0.00 seconds. 
Algebraic proofs of Dirichlet's theorem on arithmetic progressions
 Čech, Martin ; Kala, Vítězslav (advisor) ; Příhoda, Pavel (referee)
Dirichlet's theorem on arithmetic progressions says that there are infinitely many primes in any arithmetic progression an = kn + with coprime k, . The original proof of this theorem was analytic using a lot of non-elementary methods. The goal of this thesis is to give sufficient and necessary conditions on k and under which a more elementary algebraic proof of the theorem can exist, and give the proof in these cases. 1
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Counting extensions of imaginary quadratic fields
 Beneš, Alexandr ; Kala, Vítězslav (advisor) ; Yatsyna, Pavlo (referee)
The goal of this thesis is to determine the asymptotic behaviour of the number of quadratic extensions of a number field in terms of the discriminant. We will be par- ticularly interested in extensions of imaginary quadratic number fields with odd class number. For a given number field K we will define the group of ideles IK and the idele class group CK, which capture the local behaviour of a number field. Then we use the Artin reciprocity theorem to give a correspondence of quadratic extensions and quadratic characters on CK. When the class number is odd, quadratic characters on CK reduce to characters on the product of groups of units of local fields. These characters can be given explicitly and we compute the discriminant of the corresponding extension from their local conductors. We put this information together in the form of a zeta function and finally use a Tauberian theorem to compute the asymptotic behaviour. 1
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Algebraic proofs of Dirichlet's theorem on arithmetic progressions
 Čech, Martin ; Kala, Vítězslav (advisor) ; Příhoda, Pavel (referee)
Dirichlet's theorem on arithmetic progressions says that there are infinitely many primes in any arithmetic progression an = kn + with coprime k, . The original proof of this theorem was analytic using a lot of non-elementary methods. The goal of this thesis is to give sufficient and necessary conditions on k and under which a more elementary algebraic proof of the theorem can exist, and give the proof in these cases. 1
Detailed record

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