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 $L$.