Timeline for $A_5$-extension of number fields unramified everywhere

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Nov 4, 2010 at 19:21	comment	added	Pete L. Clark		@Kevin: this certainly sounds good to me. Indeed, the basic technique (choosing global extensions which have prescribed behavior at finitely many places) is one that I have used many times in my own work...
Nov 4, 2010 at 13:13	comment	added	Olivier		If I understand correctly, the approach of Yamammoto (in the specific case of $G=A_{n}$) is somewhat dual to what you suggest. Instead of killing the ramification by composing with an auxiliary extension, he realizes $G$ as a normal subgroup of a larger group $G'$ and ensures that the ramification can only occur in the quotient $G'/G$. When $G=A_{n}$, this works nicely because $G'=S_{n}$ and it is easy to build polynomials with at most one double roots modulo $p$ for all $p$.
Nov 4, 2010 at 11:42	comment	added	Kevin Buzzard		Here's an explicit worked example in a much easier case. Consider $K=\mathbf{Q}(\sqrt{2})/\mathbf{Q}$. It's the splitting field of $x^2-2$. It's ramified at 2. So let's choose a polynomial highly congruent to $x^2-2$, for example $x^2-34$, and set $L=\mathbf{Q}(\sqrt{34})$. Now $KL/L$ should be unramified everywhere and if my calculations are right, it is.
Nov 4, 2010 at 11:08	history	answered	Kevin Buzzard	CC BY-SA 2.5