Let k be a number field, Ok its ring of integers, and f(t,X) ∈ k(t)[X] be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations t 7→ t̄ ∈ Ok such that f(t̄, X) is still irreducible. In this paper we study the set Redf (Ok) of those t̄ ∈ Ok with f(t̄, X) reducible. We show that Redf (Ok) is a finite set under rather weak assumptions. In particular...

Let C be a smooth projective irreducible curve over an algebraic closed field k of characteristic 0. We consider Brill-Noether loci over the moduli space of morphisms from C to a Grassmannian G(m, n) of m−planes in k and the corresponding Quot schemes of quotients of a trivial vector bundle on C compactifying the spaces of morphisms. We study in detail the case in which m = 2, n = 4. We prove r...

4 Introduction to Galois Theory 2 4.1 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . 2 4.2 Gauss’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.3 Eisenstein’s Irreducibility Criterion . . . . . . . . . . . . . . . 6 4.4 Field Extensions and the Tower Law . . . . . . . . . . . . . . 6 4.5 Algebraic Field Extensions . . . . . . . . . . . . . . . . . . . . 8 4....

Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a weak version of rectangle condition for irreducibility of Heegaard splittings. As an application, we give an example of an irreducible genus three Heegaard splitting of a 3-manifold which is not a minimal genus Heegaard splitting.

We study the problem of constructing and enumerating, for any integers m,n > 1, number fields of degree n whose ideal class groups have “large” m-rank. Our technique relies fundamentally on Hilbert’s irreducibility theorem and results on integral points of bounded degree on curves.

We describe an explicit version of Hilbert’s irreducibility theorem using a generalization of Gallagher’s larger sieve. We give applications to the Galois theory of random polynomials, and to the images of the adelic representation associated to elliptic curves varying in rational families.

In the study of Galois theory, after computing a few Galois groups of a given field, it is very natural to ask the question of whether or not every finite group can appear as a Galois group for that particular field. This question was first studied in depth by David Hilbert, and since then it has become known as the Inverse Galois Problem. It is usually posed as which groups appear as Galois ex...

let $k$ be a field of characteristic$p>0$, $k[[x]]$, the ring of formal power series over $ k$,$k((x))$, the quotient field of $ k[[x]]$, and $ k(x)$ the fieldof rational functions over $k$. we shall give somecharacterizations of an algebraic function $fin k((x))$ over $k$.let $l$ be a field of characteristic zero. the power series $finl[[x]]$ is called differentially algebraic, if it satisfies...

In this paper, we study irreducibility of Kuramoto-Sivashinsky equation which is driven by an additive noise acting only on a finite number Fourier modes. order to obtain the irreducibility, first investigate approximate controllability finite-dimensional force, proof based Agrachev-Sarychev type geometric control approach. Next, continuity solving operator for deterministic equation. Finally, ...

We determine classes of degrees where testing irreducibility for univariate polynomials over finite fields can be done without any GCD computation. This work was partly funded by the INRIA Associate Team “Algorithms, Numbers, Computers” (http://www.loria.fr/~zimmerma/anc.html). Key-words: finite fields, irreducible polynomial, GCD ∗ ANU, Canberra, [email protected] Test d’irréductibilité de polynôme...