Although the question was only about unramified $\mathfrak{A}_5$-extensions, and has been completely answered, it might not be superfluous to mention the following paper which I happened to come across today:
MR0819826 (87e:11122)
Elstrodt, J.(D-MUNS); Grunewald, F.(D-BONN); Mennicke, J.(D-BLF)
On unramified $A_m$-extensions of quadratic number fields.
Glasgow Math. J. 27 (1985), 31–37.
An explicit description is given of unramified extensions $S/k$ with Galois group equal to the alternating group $A_n$, where $k$ is a quadratic number field. The authors prove that if $f(x)\in {\bf Z}[x]$ is a monic, irreducible polynomial of degree $n$ with square-free discriminant and Galois group $S_n$, then $S/k$ is an unramified $A_n$-extension. Here $S$ denotes the splitting field for $f(x)$ over ${\bf Q}$ and $k={\bf Q}(\sqrt{\Delta})$, where $\Delta$ is the determinant of $f$. The proof involves a series of calculations which show that $S/k$ has relative different 1.
In the final section, 84 examples of unramified $A_5$-extensions of quadratic fields are given. In 15 of the cases the quadratic field is real and in 69 cases it is imaginary. This list contains an example (with real quadratic field) due to E. Artin, which was mentioned by S. Lang [Algebraic number theory, see p. 121, Addison-Wesley, Reading, Mass., 1970].
Reviewed by Charles J. Parry
Addendum (2011/03/30) See also the recent Kedlaya's preprint mentioned by Kiran KedlayaSpeyer is now available on the arXiv. A corollary is that for each $n\geq3$, infinitely many quadratic number fields admit everywhere unramified degree-$n$ extensions whose normal closures have Galois group $\mathfrak{A}_n$.
