Related papers: Specialization results and ramification conditions

Cyclic, ramified extensions $L/K$ of degree $p$ of local fields with residue characteristic $p$ are fairly well understood. Unless $\mbox{char}(K)=0$ and $L=K(\sqrt[p]{\pi_K})$ for some prime element $\pi_K\in K$, they are defined by an…

Given a number field $F$, a finite group $G$ and an indeterminate $T$, {\it{a $G$-parametric extension over $F$}} is a finite Galois extension $E/F(T)$ with Galois group $G$ and $E/F$ regular that has all the Galois extensions of $F$ with…

The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group $G$, let $m(G)$ be the minimal integer $m$ for which there exists a Galois extension $N/\mathbb{Q}$…

Given a field $k$ and a finite group $H$, {\it{an $H$-parametric extension over $k$}} is a finite Galois extension of $k(T)$ of Galois group containing $H$ which is regular over $k$ and has all the Galois extensions of $k$ of group $H$…

The theory of p-ramification, regarding the Galois group of the maximal pro-p-extension of a number field K, unramified outside p and $\infty$, is well known including numerical experiments with PARI/GP programs. The case of ``incomplete…

We construct explicitly APF extensions of finite extensions of $\qp$ for which the Galois group is not a p-adic Lie group and which do not have any open subgroup with $\zp$-quotient.

Let $L/K$ be a finite Galois, totally ramified $p$-extension of complete local fields with perfect residue fields of characteristic $p>0$. In this paper, we give conditions, valid for any Galois $p$-group $G={Gal}(L/K)$ (abelian or not) and…

We investigate unramified extensions of number fields with prescribed solvable Galois group and certain extra conditions. In particular, we are interested in the minimal degree of a number field $K$, Galois over $\mathbb{Q}$, such that $K$…

Given a natural number n and a number field K, we show the existence of an integer \ell_0 such that for any prime number \ell\geq \ell_0, there exists a finite extension F/K, unramified in all places above \ell, together with a principally…

Given a $2$-adic field $K$, we give formulae for the number of totally ramified quartic field extensions $L/K$ with a given discriminant valuation and Galois closure group. We use these formulae to prove a refinement of Serre's mass…

Let $L/K$ be a finite Galois extension of local fields. The Hasse-Arf theorem says that if Gal$(L/K)$ is abelian then the upper ramification breaks of $L/K$ must be integers. We prove the following converse to the Hasse-Arf theorem: Let $G$…

Suppose G is a semi-direct product of the form Z/p^n \rtimes Z/m where p is prime and m is relatively prime to p. Suppose K is a local field of characteristic p > 0. The main result states necessary and sufficient conditions on the…

We introduce and study a class of field extensions that we call pre-Galois; viz. extensions that become Galois after some linearly disjoint Galois base change. Among them are geometrically Galois extensions of k(T), with k a field:…

Let $K/\mathbb{Q}$ be a finitely generated field of characteristic zero and $X/K$ a smooth projective variety. Fix $q\in\mathbb{N}$. For every prime number $\ell$ let $\rho_\ell$ be the representation of $\mathrm{Gal}(K)$ on the \'etale…

We provide evidence for this conclusion: given a finite Galois cover $f: X \rightarrow \mathbb{P}^1_\mathbb{Q}$ of group $G$, almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$. We…

Let $K$ be a local field of characteristic $p>0$ with perfect residue field and let $G$ be a finite $p$-group. In this paper we use Saltman's construction of a generic $G$-extension of rings of characteristic $p$ to construct totally…

Let $p$ be an odd prime and $F$ be a number field whose $p$-class group is cyclic. Let $F_{\{\mathfrak{q}\}}$ be the maximal pro-$p$ extension of $F$ which is unramified outside a single non-$p$-adic prime ideal $\mathfrak{q}$ of $F$. In…

The paper is concerned with the following version of Hilbert's irreducibility theorem: if $\pi: X \to Y$ is a Galois $G$-covering of varieties over a number field $k$ and $H \subset G$ is a subgroup, then for all sufficiently large and…

Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of degree d. The polynomial P_n(x,t)= f^n(x) - t in F[x] where f^n is the n-fold iterate of f, is absolutely irreducible over F; we compute a recursion for its…

In this paper, we consider infinite Galois extensions of number fields and study the relation between their local degrees and the structure of their Galois groups. It is known that, if $K$ is a number field and $L/K$ is an infinite Galois…