Related papers: Rubik's as a Galois'

For each odd prime $p$, we prove the existence of infinitely many real quadratic fields which are $p$-rational. Explicit imaginary and real bi-quadratic $p$-rational fields are also given for each prime $p$. Using a recent method developed…

Let k be a number field and K/k Galois. We transform the construction of the unramified Brauer group of the norm one torus R^1_K/k(G_m) into the construction of a special abelian extension over K. If k=Q and K/Q biquadratic, we explicitly…

We construct motivic $\ell$-adic representations of $\GQ$ into exceptional groups of type $E_7,E_8$ and $G_2$ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and uses the…

Which groups can occur as the group of units in a ring? Such groups are called realizable. Though the realizable members of several classes of groups have been determined (e.g., cyclic, odd order, alternating, symmetric, finite simple,…

This paper is on the inverse parameterized differential Galois problem. We show that surprisingly many groups do not occur as parameterized differential Galois groups over K(x) even when K is algebraically closed. We then combine the method…

We study relativized Lascar groups, which are formed by relativizing Lascar groups to the solution set of a partial type $\Sigma$. We introduce the notion of a Lascar tuple for $\Sigma$ and by considering the space of types over a Lascar…

We present a simple method for the construction of polynomials with cyclic Galois groups, hoping to encourage a reader with some background in algebra to make computations of his/her own.

In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve and let J(C) be the associated Jacobian variety. Assume that there exists a prime p such that J(C)…

We compute the Galois groups for a certain class of polynomials over the the field of rational numbers that was introduced by S. Mori and study the monodromy of corresponding hyperelliptic jacobians.

In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-$1$ subset. Our…

For a rational prime $p\neq 2$, we compute the sequence of ramification groups of a Galois, radical and finite extension $L/F$ where $F/\mathbb{Q}_p$ is an unramified finite extension. First, we compute it in the case where the exponent of…

We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such…

Let $C \subset \mathbb{P}^2$ be a plane curve of degree at least three. A point $P$ in projective plane is said to be Galois if the function field extension induced by the projection $\pi_P: C \dashrightarrow \mathbb P^1$ from $P$ is…

Let $k$ be any field, $G$ be a finite group acting on the rational function field $k(x_g:g\in G)$ by $h\cdot x_g=x_{hg}$ for any $h,g\in G$. Define $k(G)=k(x_g:g\in G)^G$. Noether's problem asks whether $k(G)$ is rational (= purely…

In this work, we establish two main results in the context of arithmetic and geometric properties of plane curves. First, we construct numerous new examples of arithmetic Zariski pairs and multiplets, where only a few ones were previously…

We extend Kovacic's algorithm to compute the differential Galois group of some second order parameterized linear differential equation. In the case where no Liouvillian solutions could be found, we give a necessary and sufficient condition…

I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number…

In this article, the Galois groupoid of the first Painlev\'{e} equation is computed. This computation use E. Cartan's classification of structural equations of pseudogroups acting on $C^2$ and the degeneration of the first Painlev\'{e}…

We prove rationality of the quotient $\mathbb{C}^n / H_n$ for the finite Heisenberg group $H_n$, any $n \ge 1$, acting on $\mathbb{C}^n$ via its irreducible representation.

This article computes the Galois groups of congruence covers arising in the context of certain hyperbolic triangle groups. As a consequence of this computation, the genera of the respective curves are deduced.