We study the compositum $k^{\left[d\right]}$ of all degree $d$ extensions of a number field $k$ in a fixed algebraic closure. We show $k^{\left[d\right]}$ contains all subextensions of degree less than $d$ if and only if $d\le 4$. We prove that for $d\>2$ there is no bound $c=c\left(d\right)$ on the degree of elements required to generate finite subextensions of $k^{\left[d\right]}/k$. Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of $d$, but that one can take $c=d$ when $d$ is prime. This question was inspired by work of Bombieri and...

We consider the polynomial $f_k\left(z\right)=z^k-z^{k-1}-\cdots -z-1$ for $k\ge 2$ which arises as the characteristic polynomial of the $k$-generalized Fibonacci sequence. In this short paper, we give estimates for the absolute values of the roots of $f_k\left(z\right)$ which lie inside the unit disk.

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed $\alpha \in \mathbb{Q}-{\mathbb{Z}}_{\<0}$, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre Polynomial $L_n^{\left(\alpha \right)}\left(x\right)={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{n+\alpha }{n-j}\right){(-x)}^j/j!$ is irreducible for all large enough $n$. We use our criterion to show that, under these conditions, the Galois group of $L_n^{\left(\alpha \right)}\left(x\right)$ is either the alternating or symmetric group on $n$ letters, generalizing results of Schur for $\alpha =0,1,\pm \frac{1}{2},-1-n$.

Let $E$ be a CM number field, $p$ an odd prime totally split in $E$, and let $X$ be the $p$-adic analytic space parameterizing the isomorphism classes of $3$-dimensional semisimple $p$-adic representations of $\mathrm{Gal}(\overline{E}/E)$ satisfying a selfduality condition “of type $\mathrm{U}\left(3\right)$”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in $X$ has dimension at least $3[E:\mathbb{Q}]$. As important steps, and in any rank, we prove that any first order...

The purpose of this paper is to prove that the Pythagorean hull of Q is the splitting field in R of a special set of rational polynomials.