How to find Galois extension given the Galois group?

Question

I want to be able to understand the basic logic on how to find Galois extension $E$ given the Galois group of this extension $E/\mathbb{Q}$, i can do this for very simple groups but i get confused at more complex ones. For example given $\mathbb{Z}_2$ i can see that the roots of the minimal polynomial alternate by a plus minus sign so the extension must be one on of $\mathbb{Q}(\sqrt{2})$, $\mathbb{Q}(\sqrt{3})$ and so on. But what is the basic idea to work on such problems, for example for $\mathbb{Z}_5$ ?