Explanations for the Hilbert symbols Are there some elementary sources which help me to understand the Hilbert symbol and the proof of the Hasse-Minkowski-theorem?
If you know anything which explains it well (except J.-P. Serre's book!) could you share it with me. Thank you!
 A: Serre's exposition is the best in my opinion and explains everything very well. If your problem is his treatment of the p-adics then I suggest you look at e.g. Gouvea for a more concrete description of them or that you take Theorems 3 and 4 in Serre's book in Chapter II as a black box. Especially from the Hilbert symbol onwards it is mainly computational. Incredible clever of course and not possible for a mortal to come up with but not too hard to follow. This is an intrinsically difficult theorem so it is not to be expected that there are truly "elementary" sources.
A: Adam Gamzon's senior thesis here proves the Hasse-Minkowski theorem over $\mathbf Q$ and $F(T)$ for finite fields $F$ of odd characteristic.  It has more examples than in Serre's account, and it approaches the Hilbert symbol in a different way: it proves the Hilbert symbol is bimultiplicative before working out a formula for the Hilbert symbol (in completions of $\mathbf Q$ other than $\mathbf Q_2$; the formula in the $2$-adic case is not needed in Gamzon's approach to the Hasse-Minkowski theorem).
You could also look at Tim Curry's senior thesis here for an account of the Hilbert symbol on completions of $\mathbf Q$ and $\mathbf Q(i)$. Here too, the bimultiplicativity of the symbol is proved before looking at formulas for it.
