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The arboreal Galois group of a polynomial $f$ over a field $K$ encodes the action of Galois on the iterated preimages of a root point $x_0in K$, analogous to the action of Galois on the $ell$-power torsion of an abelian variety. We compute the arboreal Galois group of the postcritically finite polynomial $f(z) = z^2 - 1$ when the field $K$ and root point $x_0$ satisfy a simple condition. We call the resulting group the arithmetic basilica group because of its relation to the basilica group associated with the complex dynamics of $f$. For $K=mathbb{Q}$, our condition holds for infinitely many choices of $x_0$.
We formulate a general question regarding the size of the iterated Galois groups associated to an algebraic dynamical system and then we discuss some special cases of our question.
We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of their Galois groups as subgroups of automorphism gr
In this paper we show an explicit polynomial in Q[x] that has Galois group SL2(F16), filling in a gap in the tables of Juergen Klueners and Gunther Malle. The computation of this polynomial uses modular forms and their Galois representations.
Let $ksubseteq K$ be a finite Galois extension of fields with Galois group $G$. Let $mathscr{G}$ be the automorphism $k$-group scheme of $K$. We construct a canonical $k$-subgroup scheme $underline{G}subsetmathscr{G}$ with the property that $Spec_k(K
For positive integers $n$, the truncated binomial expansions of $(1+x)^n$ which consist of all the terms of degree $le r$ where $1 le r le n-2$ appear always to be irreducible. For fixed $r$ and $n$ sufficiently large, this is known to be the case. W