NMAG470 The seminar is intended for students, faculty and occasional visitors to give accessible talks on various areas of number theory.
Depending on the participants, the talks will be in Czech or English. If you're interested in attending the seminar or giving a talk, please email me ( We have a mailing list for announcing the upcoming talks (and occasional other related things). Please email me if you want to be added to the list.
In 2018-2019 the seminar is co-organized by Magda Tinková and Tomáš Vávra. Schedulesummer break till October 201921. 5. Nihal Bircan Kaya (Cankiri, Turkey), On Sequences of Integers of Quadratic Fields and Relations with Artin's Primitive Root Conjecture I will consider the integers alpha of the quadratic field Q(sqrt d) where d is a square-free integer. Using the embedding into GL(2,R) we obtain bounds for the first n such that alpha^n = 1 mod p. More generally, if O_f is a number ring of conductor f, we study the first integer n=n(f) such that alpha^n lies in O_f. We obtain bounds for n(f) and for n(fp^k). We allow any norm N(alpha)<>0. The case where alpha is the fundamental unit in a real quadratic number field is of special interest. We also study a certain probability distribution suggested by the numerical results. In the second part of my talk I will indicate in details how my results relate to Artin primitive root type problems over quadratic fields. 9. 5. at 15:40 in K12, Marine Rougnant (Besançon), p-rationality of number fields of low degree A prime in a Galois extension can remain prime or not, and such
information is governed by ramification theory. Specifically, for a
fixed prime p, we can consider the maximal extension unramified outside
p. The Galois group of this extension is linked to the conjecture of
Gras on p-rational fields.We propose to
explore this conjecture in the case of some low degree fields: first
theoretically, using the abc conjecture, then experimentally, with
PARI/GP computations. We will see that these two points of view support
the conjecture.30. 4. Jakub Krásenský, Universal quadratic forms over biquadratic fields 23. 4. Kristína Mišlanová, Matice Legendreových symbolov [Matrix of Legendre symbols, in Slovak] Pri
skúmaní znamienkových matíc Dummit-Dummit-Kisilevsky nedávno definovali
jednu ich triedu ako matice Legendreových symbolov modulo rôzne
prvočísla (matice kvadratických zvyškov) a aj ich zovšeobecnenie v
podobe matíc kubických zvyškov definovaných pomocou kubických mocninných
symbolov. Na základe tohto článku ukážeme ako sú tieto matice
definované a popíšeme ich charakterizáciu, predovšetkým blokový tvar
týchto matíc. Ďalším krokom je zovšeobecnenie tejto charakterizácie pri
rozšírení definícii na voľbu neprimárnych prvočiniteľov.17. 4. at 15:40 in KA (seminar KAFKA): Víťa Kala, Lifting problem for universal quadratic forms 16. 4. Tomáš Hejda, Ternary universal quadratic forms in arithmetic sequences We say that a diagonal ternary positive form Q(x,y,z)=ax^2+by^2+cz^2 over Z is (k,l)-universal (for positive k and l with l<k) iff all natural numbers of the form n=km+l are represented by Q. We say that Q is almost (k,l)-universal iff all but finitely many such numbers are represented. We show that almost universal forms exist for all k and non-zero l. Then we restrict to the case k=p is prime and we give numerical and statistical arguments why one should conjecture whether there are finitely or infinitely many (p,l)-universal forms. 2. 4. 2019 at 14:00 in K5: Hartmut Monien (Bonn), Inverse Galois theory: new results for genus zero sporadic groups 2. 4. 2019: Matěj Doležálek (Humpolec), Quaternions, four-square theorems and their analogues Lagrange's
four-square theorem states that any positive integer can be represented
as the sum of four squares of integers, whereas Jacobi's four-square
theorem gives the exact number of such representations for a given
positive integer. In this talk, we will show how one can prove these
theorems by examining some algebraic and ring-theoretic properties of
quaternions, as well as how this approach can be modified to prove the
analogous theorems for certain other quadratic forms.In
1839 Hermite posed to Jacobi the problem of finding a method for
representing real numbers by sequences of nonnegative integers, such
that the periodic representations would correspond to the algebraic
properties of the numbers (especially to the degree). Continued
fractions completely solve this problem for quadratic irrationalities,
but for numbers of degree >2 it showed to be a very hard problem.
Starting with Jacobi, there were published many modifications of the
classical continued fraction algorithm, called multidimensional
continued fractions, that attempts to solve this question.In this talk we give a summary of the vectorial multidimensional continued fractions and its algebraic properties.19. 3. 2019: Magdaléna Tinková, Indecomposable integers in real quadratic fields of odd discriminant In
this talk, we will discuss so-called indecomposable integers in real
quadratic fields. In 2016, Jang and Kim stated a conjecture about these
elements, which, in some cases, was later disproved by Kala. We will
show some results related to this conjecture and briefly sketch the
method which leads to finding counterexamples in all the cases of
quadratic fields. This is joint work with Paul Voutier.12. 3. 2019: Kristýna Zemková (Dortmund), Non-normal number fields Considering the cyclotomic number field Q(zeta_k) (where zeta_k
is a primitive k-th root of unity), what can be said about its
quadratic extensions? In particular, we are interested in the extensions
of the form Q(sqrt{a+bzeta_k}). For
certain pairs of integers a, b and k>2 (k<>6), we show that
such a field is not normal over Q. (Joint work with Pieter
Moree and Carlo Pagano.) 8. 3. 2019: Prague-Dresden Number Theory Day 26. 2. 2019: Ezra Waxman, Analytic number theory in function fields Let denote the finite field with elements, and let denote the ring of polynomials with coefficients in . One of the guiding principles in modern number theory is the deep connection between function fields (such as , the fraction field of ) and number fields (such as the field of rational numbers, ). In particular, the statistical behavior of prime polynomials in the ring mirrors
that of the ordinary prime numbers. In this talk, I will elaborate on
this rich analogy between function fields and number fields. In
particular, I will provide a proof of the prime polynomial theorem (i.e.
the function field analogue of the prime number theorem), and discuss
several open problems related to the statistical distribution of prime
numbers that have been successful resolved in the function field
setting. |

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