We study the monodromy groups of compositions of two indecomposable polynomials. In particular, we show that such monodromy groups either fulfill a certain "largeness" property, or are in an explicit list of exceptions. Such largeness results are crucial for dealing with compositions of more than two polynomials, and consequently are expected to have a wide range of applications to problems concerning the arithmetic of polynomials. Concretely, our main result is a key ingredient in the solution of a long-standing open problem due to Davenport, Lewis and Schinzel, achieved in a companion paper.

We solve the problem of Davenport–Lewis–Schinzel (DLS), originating in the 1950s, regarding the reducibility of f(X)-g(Y)∈ℂ[X,Y]. This yields an almost-complete solution to the Hilbert–Siegel problem: For a polynomial map f whose composition factors avoid only very specific low-degree polynomials, we explicitly describe over which integers the fibers of f are reducible. We further apply the solution to stability of iterates of f in arithmetic dynamics, and to solving the functional equation f(X)=g(Y) in X,Y∈ℂ(z).

We give an affirmative answer to the Grunwald problem for new families of non-solvable finite groups G, away from the set of primes dividing |G|. Furthermore, we show that such G verify the condition (BM), that is, the Brauer-Manin obstruction to weak approximation is the only one for quotients of SL_n by G. These new families include extensions of groups satisfying (BM) by kernels which are products of symmetric groups S_m, with m≠ 2,6, and alternating groups A_5. We also investigate (BM) for small groups by giving an explicit list of small order groups for which (BM) is unknown and we show that for many of them (BM) holds under Schinzel's hypothesis.

Given two G-Galois extensions of ℚ, is there an extension of ℚ(t) that specializes to both? The equivalence relation on G-Galois extension of ℚ, induced by the above question, is called R-equivalence. The number of R-equivlance classes indicates how many rational spaces are required in order to parametrize all G-Galois extensions of ℚ. We determine the R-equivalence classes for basic families of groups G, and consequently obtain parametrizations of the G-Galois extensions of ℚ in the absence of a generic extension for G.

For a composition f=f_1∘⋯∘ f_r of polynomials f_i∈ℚ[x] of degrees d_i≥ 5 with alternating or symmetric monodromy group, we show that the monodromy group of f contains the iterated wreath product A_d_r≀⋯≀ A_d_1. A similar property holds more generally for polynomials that do not factor through x^d or Chebyshev. We derive consequences to arithmetic dynamics regarding arboreal representations, and forward and backward orbits of such f. In particular, given an orbit (a_n)_n=0^∞ of f as above, we show that for "almost all" a∈ℤ, the set of primes p for which some a_n is congruent to a mod p is "small".

For each nonnegative integer g, we classify the ramification types and monodromy groups of indecomposable coverings of complex curves f: X→ Y where X has genus g, under the hypothesis that n:=(f) is sufficiently large and the monodromy group is not A_n or S_n. This proves a conjecture of Guralnick and several conjectures of Guralnick and Shareshian.

The combination of this paper and its companion complete the classification of monodromy groups of indecomposable coverings of complex curves f:X→ℙ^1 of sufficiently large degree in comparison to the genus of X. In this paper we determine all such coverings with monodromy group G≤ S_ℓ≀ S_t of product type for t≥ 2.

For a degree $n$ polynomial $f\in {\mathbb {Q}}[x]$, the elements in the fiber $f<^>{-1}(a)\subseteq {\mathbb {C}}$ are of degree $n$ over ${\mathbb {Q}}$ for most values $a\in {\mathbb {Q}}$ by Hilbert's irreducibility theorem. Determining the set of exceptional $a$'s without this property is a long standing open problem that is closely related to the Davenport-Lewis-Schinzel problem (1959) on reducibility of variable separated polynomials. As opposed to a previous work that mostly concerns indecomposable $f$, we answer both problems for decomposable $f=f_{1}\circ \cdots \circ f_{r}$, as long as the indecomposable factors $f_{i}\in {\mathbb {Q}}[x]$ are of degree $\geq 5$ and are not $x<^>{n}$ or a Chebyshev polynomial composed with linear polynomials.

By a classical result of Neukirch, Ikeda, Iwaswa and Uchida, a number field K is determined by the structure of its absolute Galois group Gal(K). We show that K is not determined by the structure of the Sylow subgroups of Gal(K), answering a question raised by Florian Pop.

A finite group G is admissible over a field M if there is a division algebra whose center is M with a maximal subfield G-Galois over M. We consider nine possible notions of being admissible over M with respect to a subfield K of M, where the division algebra, the maximal subfield or the Galois group are asserted to be defined over K. We completely determine the logical implications between all variants.

Given a finite group G and a number field K , we investigate the following question: Does there exist a Galois extension E/K ( t ) with group G whose set of specializations yields solutions to all Grunwald problems for the group G , outside a finite set of primes? Following previous work, such a Galois extension would be said to have the “Hilbert–Grunwald property”. In this paper we reach a complete classification of groups G which admit an extension with the Hilbert–Grunwald property over fields such as K = ℚ. We thereby also complete the determination of the “local dimension” of finite groups over ℚ.

Let G G be a finite group and K K a number field. We construct a G G -extension E / F E/F , with F F of transcendence degree 2 2 over K K , that specializes to all G G -extensions of K p K_\mathfrak {p} , where p \mathfrak {p} runs over all but finitely many primes of K K . If furthermore G G has a generic extension over K K , we show that the extension E / F E/F has the so-called Hilbert–Grunwald property. These results are compared to the notion of essential dimension of G G over K K , and its arithmetic analogue.

Given an irreducible bivariate polynomial f (t, x) ∈ ℚ[t, x], what groups H appear as the Galois group of f (t0, x) for infinitely many t0 ∈ ℚ? How often does a group H as above appear as the Galois group of f(t0, x), t0 ∈ ℚ? We give an answer for f of large x-degree with alternating or symmetric Galois group over ℚ(t). This is done by determining the low genus subcovers of coverings $$\tilde X \to \mathbb{P}_\mathbb{C}^1$$ with alternating or symmetric monodromy groups.

The finite subgroups of $$\mathrm{PGL}_2({\mathbb {C}})$$ are shown to be the only finite groups G with this property: for some integer $$r_0$$ (depending on G), all Galois covers $$X\rightarrow {\mathbb {P}}^1_{\mathbb {C}}$$ of group G can be obtained by pulling back those with at most $$r_0$$ branch points along non-constant rational maps $${\mathbb {P}}^1_{\mathbb {C}}\rightarrow {\mathbb {P}}^1_{\mathbb {C}}$$ . For $$G\subset \mathrm{PGL}_2({\mathbb {C}})$$ , it is in fact enough to pull back one well-chosen cover with at most 3 branch points. A consequence of the converse for inverse Galois theory is that, for $$G\not \subset \mathrm{PGL}_2({\mathbb {C}})$$ , letting the branch point number grow provides truly new Galois realizations $$F/{\mathbb {C}}(T)$$ of G. Another application is that the “Beckmann–Black” property that “any two Galois covers of $${\mathbb {P}}^1_{\mathbb {C}}$$ with the same group G are always pullbacks of another Galois cover of group G” only holds if $$G\subset \mathrm{PGL}_2({\mathbb {C}})$$ .

Given fields k⊆L, our results concern one parameter L-parametric polynomials over k, and their relation to generic polynomials. The former are polynomials P(T,Y)∈k[T][Y] of group G which parametrize all Galois extensions of L of group G via specialization of T in L, and the latter are those which are L-parametric for every field L⊇k. We show, for example, that being L-parametric with L taken to be the single field C((V))(U) is in fact sufficient for a polynomial P(T,Y)∈C[T][Y] to be generic. As a corollary, we obtain a complete list of one parameter generic polynomials over a given field of characteristic 0, complementing the classical literature on the topic. Our approach also applies to an old problem of Schinzel: subject to the Birch and Swinnerton-Dyer conjecture, we provide one parameter families of affine curves over number fields, all with a rational point, but with no rational generic point.

The only perfect powers in the Fibonacci sequence are 0, 1, 8, and 144, and in the Lucas sequence, the only perfect powers are 1 and 4. We prove that in sequences that follow the same recurrence relation of the Lucas and Fibonacci sequences, there are always only finitely many polynomial values g(Z) for any polynomial g which is not equivalent to a Dickson polynomial.

Given an irreducible polynomial F in Q(t)[x], we develop methods for determining the set of exceptions in Hilbert's irreducibility theorem up to a finite set, under nonsolvability assumptions on its Galois group A=Gal(F/Q(t)). As opposed to previous results, these methods address the case where A is imprimitive. As a consequence, we answer the Davenport-Lewis-Schinzel problem (1959), and the problem of determining the reducibility of fibers of a polynomial f in Q[x] over rational points (1971), when the involved polynomials do not factor through x^n, a Chebyshev polynomial, and an indecomposable degree 4 polynomial.

For various nonsolvable groups G, we prove the existence of extensions of the rationals Q with Galois group G and inertia groups of order dividing ge(G), where ge(G) is the smallest exponent of a generating set for G. For these groups G, this gives the existence of number fields of degree ge(G) with an unramified G-extension. The existence of such extensions over Q for all finite groups would imply that, for every finite group G, there exists a quadratic number field admitting an unramified G-extension, as was recently conjectured. We also provide further evidence for the existence of such extensions for all finite groups, by proving their existence when Q is replaced with a function field k(t) where k is an ample field.

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 crossed products, the Hilbert--Grunwald property, and finite parametric sets.

The existence of finite dimensional central division algebras with no maximal subfield that is Galois over the center (called noncrossed products), was for a time the biggest open problem in the theory of division algebras, before it was settled by Amitsur. Motivated by Brussel’s discovery of noncrossed products over Q((t)), we describe the “location” of noncrossed products in the Brauer group of general Henselian valued fields with arbitrary value group and global residue field. We show that within the fibers defined canonically by Witt’s decomposition of the Brauer group of such fields, crossed products and noncrossed products are, roughly speaking, separated by an index bound. This generalizes a result of the first and third author for rank 1 valued Henselian fields. Furthermore, we prove that all fibers which are not covered by the rank 1 case, and where the characteristic of the residue field does not interfere, contain noncrossed products. We show by example that, unlike in the rank 1 case, the value of the index bound does not depend on the number of roots of unity that are present. Thus, the index bounds are in general of a different nature than in the rank 1 case.