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Lehmer's polynomial's Galois group is a subgroup of the wreath product $C_2 \wr S_5$ simply because it's reciprocal, i.e. $\alpha$ is a root of the polynomial if and only if $\alpha^{-1}$ is. (This is easily seen to be equivalent to the polynomials coefficients' being palindromic.) You can easily see that the Galois group of a reciprocal polynomial must respect this partition of the $2n$ roots into $n$ pairs (of a number and its reciprocal). In general, if a permutation group respects this kind of block structure, it's called imprimitive and contained in a wreath product as Geoff describes.
@igor If you run computations on Galois groups of reciprocal polynomials of degree 2n, you'll find that having a "half full" Galois group is quite common. The "full" group is $C_2 \wr S_n$, corresponding to the "full permutation module" for $S_n$ over $\mathbb{F}_2$, whereas the "half full" group corresponds to the "standard module" (of dimension $n-1$). It was shown by McKee and Christopoulos that these two situations are the only ones that can happen for Salem numbers (although the $S_n$ can be replaced by a smaller transitive group of degree $n$.
added group theory tag -- as you can see in the discussions in the answers, the question is related to group theory questions phrased outside the langauge of profinite groups.
@VesselinDimitrov: as far as you know, is it possible that there exists a variety $V$ such that $V(F)$ is finite for any field $F$ with the Northcott property?
@VesselinDimitrov Actually the question of Amoroso, David, and Zannier has been settled. In this note: arxiv.org/abs/1408.6411 Lukas Pottmeyer constructs an example of a PAC field with the Bogomolov property (see the last section).
Wait -- but the bound we get this way also depends on $\ell$ (unlike in the statement of conjecture 2, 2', 2'' above), right? Specifically, the "gonality grows linearly" statement and/or the "only finitely many subgroups of a given index" statement are dependent on $\ell$, yes?
Given a number field, sage (and other things) can compute the unit group for you. I think finding good algorithms is a fairly classical problem in computational number theory, but algorithms for this are well known. You might check out Section 4.9 in Cohen's "A Course in Computational Algebraic Number Theory."
@René when you say "the argument with heights," I think of the argument going back to Bombieri & Zannier that, in a Galois extension of $\mathbb{Q}$ which sits in a finite extension of some $\mathbb{Q}_p$, the lim inf of the logarithmic height values (on $\mathbb{G_m}$) is positive. Is this what you mean?
