I recently came across the following statement, 


The Galois coverings of $\mathbb{P}^1\_\mathbb{Q}$ are all of the form
$$\mathbb{P}^1\_L\to\mathbb{P}^1\_\mathbb{Q}$$ where $L$ is a number field.


How to prove this ,


Any reference.



Any galois covering of $P^{1}$ over rationals are of the form $\mathbb{P}^1_L\to\mathbb{P}^1_\mathbb{Q}$
Let $X$ be a smooth curve over a field $K$. Then there is an exact sequence of etale fundamental groups:


$$1\rightarrow\pi\_1(X\_{\overline{K}})\rightarrow\pi\_1(X)\rightarrow\pi\_1(Spec\;K)\rightarrow 1$$


Taking $X= \mathbb{P}^1\_K$, we find that $X\_{\overline{K}}$ has trivial etale fundamental group (if $K$ is a number field, this follows from the Riemann existence theorem, and the fact that $\mathbb{P}^1\_{\mathbb{C}}$ is the Riemann sphere, which is simply connected).


Thus, the natural map $\pi\_1(X)\rightarrow\pi\_1(Spec\; K)$ is an isomorphism. If you think about the Galois correspondence, then you'll find that this means precisely that every cover of $X$ comes from pulling back a cover of Spec $K$.


If you're interested in questions like this, I would really recommend that you learn about Galois categories. There are lots of books/articles on this. For example, Lenstra's *Galois Theory for Schemes* is very readable and is available online:


<http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf>


