Unramified Alternating Extensions of Quadratic Fields

We exhibit, for each n ≥ 5, infinitely many quadratic number fields admitting unramified degree n extensions with prescribed signature whose normal closures have Galois group An. This generalizes a result of Uchida and Yamamoto, which did not include the ability to restrict the signature, and a result of Yamamura, which was the case n = 5. It is a folk conjecture that for n ≥ 5, all but finitely many quadratic number fields admit unramified extension fields of degree n whose normal closures have Galois group An, the alternating group on n symbols. Uchida [2] and Yamamoto [3] proved independently that there exist infinitely many real and infinitely many imaginary fields with unramified An-extensions. Our main theorem is the following refinement of this result; the case n = 5 was previously obtained by Yamamura [4] using a different argument. Theorem 1. For r = 0, 1, . . . , ⌊n/2⌋, there exist infinitely many real quadratic fields admitting an unramified degree n extension with Galois group An and having exactly r complex embeddings. In fact, the number of such real quadratic fields with discriminant at most N is at least O(N). Moreover, these assertions remain true if we require all fields involved to be unramified over a finite set of finite places of Q. Proof. The idea is to construct monic polynomials P (x) with integer coefficients and squarefree discriminant ∆, so that Q[x]/(P (x)) is unramified over Q( √ ∆). To do so, we construct Q(x) = n(x− a1) · · · (x− an−1) and set Pb(x) = b+ ∫ x 0 P (t) dt. Then the discriminant ∆b of Pb(x) factors as ∆b = n n n−1