$A_5$-extension of number fields unramified everywhere So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and solvable extensions unramified everywhere for his students to play with and that he had find this easy to construct with class field theory in the back of his head. But then he asked me if I knew how to construct an extension of number fields with Galois group $A_{5}$ and unramified everywhere. All I could say at the time (and now) is:


*

*There are Hilbert modular forms unramified everywhere. 

*There are Hilbert modular forms whose residual $G_{{F}_{v}}$-representation mod $p$ is trivial for all $v|p$.

*There are Hilbert modular forms whose residual $G_{F}$-representation mod $p$ has image $A_{5}$ inside $\operatorname{GL}_{2}(\mathbb F_{p})$. 


Suppose there is a Hilbert modular form satisfying all three conditions. Then the Galois extension through which its residual $G_{F}$-representation factors would have Galois group $A_{5}$ and would be unramified everywhere. 
Can this be made to work?
Regardless of the validity of this circle of idea, can you construct an extension of number fields unramified everywhere and with Galois group $A_{5}$?
 A: 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) Kedlaya's preprint mentioned by Speyer 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 $\mathfrak{A}_n$.
Addendum  Kedlaya's paper has now appeared in the Proceedings of the AMS 140 (2012), 3025--3033
A: If you take the splitting field of $x^5+ax+b$ and consider it as an extension of its quadratic subfield, then it will be unramified with Galois group contained in $A_5$ whenever $4a$ and $5b$ are relatively prime. This is a result of Yamamoto. For almost all $a$ and $b$ (specifically, on the complement of a thin set), the group is $A_5$.
You might also enjoy this preprint of Kedlaya, which I found very readable. A note on Kedlaya's webpage, dated May 2003, says that he will not be publishing this because it has been superseded by a recent result of Ellenberg and Venkatesh. I assume he is referring to this paper, but I can't figure out why that one supersedes his.
A: Here's the standard example. I found it in Lang's Algebraic Number Theory
where he attributes it to Artin. Let $K$ be the splitting field of $X^5-X+1$
over $\mathbb{Q}$. Then $K$ has Galois group $S_5$ over $\mathbb{Q}$
and $A_5$ over $L=\mathbb{Q}(\sqrt{2869})$. Also $K$ is unramified over $L$.
A: Oh, I know how I would try and build examples. First I would write down a random $A_5$ extension $K$ of $\mathbf{Q}$, ramified at some primes (in fact I would look in a table, e.g. in Buhler's thesis or the Frey et al LNM on Artin's conjecture, to find an example ramified at only one prime, probably). I would then try to kill this ramification by making an extension $L$ of $\mathbf{Q}$ which is disjoint from $K$ globally but exactly the same as it locally for the ramified primes (e.g. by going up to a splitting field of a polynomial which is highly $p$-adically congruent to the original poly whose splitting field was the $A_5$ field, for all $p$ where $K$ ramified). I'd then look at the extension $LK/L$ which I think now should be totally split at the primes above the prime where $K/\mathbf{Q}$ ramified and hence unramified everywhere. I think this stands a fair chance of working, and indeed becoming a general machine which turns an extension of number fields with group $G$ into an extension of number fields unramified everywhere with group $G$. Or have I missed something? If not then this is undoubtedly a well-known technique.
