# Number Theory Seminar 2018/19

*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.*

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.*

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.*

*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.*

26. 3. 2019: Hana Dlouhá, Algebraic properties of multidimensional continued fractions*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.)*

*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.*