On parametric and generic polynomials with one parameter

Let $k$ be an arbitrary field. We study a general method to solve the subfield problem of generic polynomials for the symmetric groups over $k$ via Tschirnhausen transformation. Based on the general result in the former part, we give an explicit solution to the field isomorphism problem and the subfield problem of cubic generic polynomials for $\frak{S}_3$ and $C_3$ over $k$. As an application of the cubic case, we also give several sextic generic polynomials over $k$.

The paper has three main applications. The first one is this Hilbert-Grunwald statement. If $f:X\rightarrow \Pp^1$ is a degree $n$ $\Qq$-cover with monodromy group $S_n$ over $\bar\Qq$, and finitely many suitably big primes $p$ are given with partitions $\{d_{p,1},..., d_{p,s_p}\}$ of $n$, there exist infinitely many specializations of $f$ at points $t_0\in \Qq$ that are degree $n$ field extensions with residue degrees $d_{p,1},..., d_{p,s_p}$ at each prescribed prime $p$. Th...

We disprove a strong form of the Regular Inverse Galois Problem: there exist finite groups $G$ which do not have a realization $F/\Qq(T)$ that induces all Galois extensions $L/\Qq(U)$ of group $G$ by specializing $T$ to $f(U) \in \Qq(U)$. For these groups, we produce two extensions $L/\Qq(U)$ that cannot be simultaneously induced, thus even disproving a weaker Lifting Property. Our examples of such groups $G$ include symmetric groups $S_n$, $n\geq 7$, infinitely many $PSL_2(\...

Let $k$ be a field of characteristic $\neq 2$. We survey a general method of the field intersection problem of generic polynomials via formal Tschirnhausen transformation. We announce some of our recent results of cubic, quartic and quintic cases the details of which are to appear elsewhere. In this note, we give an explicit answer to the problem in the cases of cubic and dihedral quintic by using multi-resolvent polynomials.

Given a field $k$ of characteristic zero and an indeterminate $T$ over $k$, we investigate the local behaviour at primes of $k$ of finite Galois extensions of $k$ arising as specializations of finite Galois extensions $E/k(T)$ (with $E/k$ regular) at points $t_0 \in \mathbb{P}^1(k)$. We provide a general result about decomposition groups at primes of $k$ in specializations, extending a fundamental result of Beckmann concerning inertia groups. We then apply our result to study...

Let $P\in\mathbb Q[t,x]$ be a polynomial in two variables with rational coefficients, and let $G$ be the Galois group of $P$ over the field $\mathbb Q(t)$. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $c$ the specialized polynomial $P(c,x)$ has Galois group isomorphic to $G$ and factors in the same way as $P$. In this paper we discuss methods for computing the group $G$ and obtaining an explicit description of the exceptional numbers $c$, i....

The aim of Bogomolov's programme is to prove birational anabelian conjectures for function fields $K|k$ of varieties of dimension $\geq 2$ over algebraically closed fields. The present article is concerned with the 1-dimensional case. While it is impossible to recover $K|k$ from its absolute Galois group alone, we prove that it can be recovered from the pair $(\mathrm{Aut}(\overline{K}|k),\mathrm{Aut}(\overline{K}|K))$, consisting of the absolute Galois group of $K$ and the l...

This survey is about Galois theory of curves in characteristic p, a topic which has inspired major research in algebraic geometry and number theory and which contains many open questions. We illustrate important phenomena which occur for covers of curves in characteristic p. We explain key results on the structure of fundamental groups. We end by describing areas of active research and giving two new results about the genus and p-rank of certain covers of the affine line.

Given a field $k$ of characteristic zero and an indeterminate $T$, the main topic of the paper is the construction of specializations of any given finite extension of $k(T)$ of degree $n$ that are degree $n$ field extensions of $k$ with specified local behavior at any given finite set of primes of $k$. First, we give a full non-Galois analog of a result with a ramified type conclusion from a preceding paper and next we prove a unifying statement which combines our results and...

Given a number field $k$, we show that, for many finite groups $G$, all the Galois extensions of $k$ with Galois group $G$ cannot be obtained by specializing any given finitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$ regular. Our examples include abelian groups, dihedral groups, symmetric groups, general linear groups over finite fields, etc. We also provide a similar conclusion while specializing any given infinitely many Galois extensions $E/k(T)$ wi...