In an old exam of my Galois Theory class there is the following question which troubles me:
Let $p \neq 2$ be a prime number and $k \geq 1$ an integer. Give an example of a galois extension $L/K$ such that $Gal(L/K) = D_{2p^k}$ and $[K:\mathbb{Q}]<+\infty$.
My idea was to consider the $p^k$-th roots of unity on which $D_{2p^k}$ acts and then take $L=\mathbb{Q}(\mu _{p^k})$ and $K=\mathbb{Q}(\mu _{p^k})^{D_{2p^k}}$ which by (what we called in class) Artin's theorem would give us $Gal(L/K) = D_{2p^k}$.
What troubles me is that I want to stop here and say that I'm done but I haven't use the $p^k,p\neq2$ conditions (what i did would work all the same with $D_{2n}$ for all $n$) so I feel that I have very probably made a mistake.