Related papers: An introduction to $\Gamma$-number fields

Let $K$ be a number field of degree at least $3$. In this article we show that the genus of the integral trace form of $K$ contains only one spinor genus. Additionally we show that exactly $43%$ (resp. $29%$, resp. $58%$) of quadratic…

In the mid 80's Conner and Perlis showed that for cyclic number fields of prime degree $p$ the isometry class of integral trace is completely determined by the discriminant. Here we generalize their result to tame cyclic number fields of…

Let $K$ be a number field. The \textit{integral trace form} is the integral quadratic form given by $\text{tr}_{K/\mathbb{Q}}(x^2)|_{O_{K}}.$ In this article we study the existence of non-conjugated number fields with equivalent integral…

Let $m>1$ and $\mathfrak{d} \neq 0$ be integers such that $v_{p}(\mathfrak{d}) \neq m$ for any prime $p$. We construct a matrix $A(\mathfrak{d})$ of size $(m-1) \times (m-1)$ depending on only of $\mathfrak{d}$ with the following property:…

In the past the first named author has studied to what extent the integral trace can characterize a number field beyond what the discriminant does. The cases of cyclic number fields and non-totally real fields are more or less settled,…

Let $K$ be a number field, which is tame and non totally real. In this article we give a numerical criterion, depending only on the ramification behavior of ramified primes in $K$, to decide whether or not the integral trace of $K$ is…

Let $N$ be a positive integer and $\Gamma$ be a subgroup of $\mathrm{SL}_2(\mathbb{Z})$ containing $\Gamma_1(N)$. Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order of discriminant $D_\mathcal{O}$ in $K$. Under some…

For every odd integer $n \geq 3$, we prove that there exist infinitely many number fields of degree $n$ and associated Galois group $S_n$ whose class number is odd. To do so, we study the class groups of families of number fields of degree…

For an abelian number field of odd degree, we study the structure of its 2-Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow…

A number field $k$ admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is bigger than one. In this case, there are only finitely many classes of such binary…

The propagation of scalar and spinor fields in a spacetime whose metric changes signature is analyzed. Recent work of Dray et al. on particle production from signature change for a (massless) scalar field is reviewed, and an attempt is made…

We compute the spinor class field for a genus of orders, in a central simple algebra of higher dimension, that are intersections of two maximal orders. In particular, we compute the number of spinor genera in a genus of such orders, as the…

Let n be an odd number and F an imaginary quadratic field with odd discriminant. We show that there exists infinitely many cubic fields K such that the class number of K is divisible by n and the Galois closure of K contains F.

Let $D$ be a totally definite quaternion algebra over a totally real number field $F$, and $\mathcal{O}$ be an $O_F$-order (of full rank) in $D$. The type number $t(\mathcal{O})$ is an important arithmetic invariant of $\mathcal{O}$ that…

We construct an infinite family of real cyclotomic fields with non-trivial class group. This result generalizes the result in [1] in the sense that our family includes theirs.

Given a nonzero integer $d$, we know by Hermite's Theorem that there exist only finitely many cubic number fields of discriminant $d$. However, it can happen that two non-isomorphic cubic fields have the same discriminant. It is thus…

In this thesis, we develop algorithms similar to the Gaussian elimination algorithm in symplectic and split orthogonal similitude groups. As an application to this algorithm, we compute the spinor norm for split orthogonal groups. Also, we…

Let $K$ be a 2-dimensional global field of characteristic $\neq 2$, and let $V$ be a divisorial set of places of $K$. We show that for a given $n \geqslant 5$, the set of $K$-isomorphism classes of spinor groups $G = \mathrm{Spin}_n(q)$ of…

For a prime \(p\ge 2\) and a number field K with p-class group of type (p,p) it is shown that the class, coclass, and further invariants of the metabelian Galois group \(G=Gal(F_p^2(K) | K)\) of the second Hilbert p-class field \(F_p^2(K)\)…

Let $p$ be an irregular prime. Let $K=\Q(\zeta)$ be the $p$-cyclotomic field. From Kummer and class field theory, there exist Galois extensions $S/\Q$ of degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree $[S:K]=p$.…