On the Galois group of Generalized Laguerre Polynomials

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be ``large.'' For a fixed \alpha \in \Q - \Z_{<0}, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre Polynomial $L_n^{(\alpha)}(x) = \sum_{j=0}^n \binom{n+\alpha}{n-j}(-x)^j/j!$ is irreducible for all large enough $n$. We use our criterion to show that, under these conditions, the Galois group of \La is either the alternating or symmetric group on $n$ letters, generalizing results of Schur for $\alpha=0,1$.Comment: 6 page

On a theorem of Koch

We give a short proof of a slightly stronger version of a theorem of Koch: A complex quadratic field whose ideal class group contains a subgroup of type (4, 4, 4) possesses an infinite unramified Galois pro-2 extension

Finitely ramified iterated extensions

Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of degree d. The polynomial P_n(x,t)= f^n(x) - t in F[x] where f^n is the n-fold iterate of f, is absolutely irreducible over F; we compute a recursion for its discriminant. Let L=L(f) be the field obtained by adjoining to F all roots, in a fixed algebraic closure, of P_n for all n; its Galois group Gal(L/F) is the iterated monodromy group of f. The iterated extension L/F is finitely ramified if and only if f is post-critically finite (pcf). We show that, moreover, for pcf polynomials f, every specialization of L/F at t=t_0 in K is finitely ramified over K, pointing to the possibility of studying Galois groups with restricted ramification via tree representations associated to iterated monodromy groups of pcf polynomials. We discuss the wildness of ramification in some of these representations, describe prime decomposition in terms of certain finite graphs, and also give some examples of monogene number fields.Comment: 19 page

Modular forms and elliptic curves over the field of fifth roots of unity

Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F.Comment: Added appendix by Mark Watkins, who found an elliptic curve missing from our tabl