the global m-th power reciprocity law and Quartic Reciprocity Law I'm reading Cox "Primes of the form $x^2+ny^2$". And I read a chapter about the global m-th power reciprocity law. Now I'm not able to prove the quartic and cubic reciprocity laws. Where can i find the proofs of them using global m-th power reciprocity law. There is a link to Hasse's book "Bericht uber neuere Untersuchungen und Probleme as der Theorie def algebraischen Zahlkorper" in Cox's book. But I haven't found English translation of it. Any recommendations for such books or articles would be of great utility. 
@Dietrich Burde
I read the chapter about Cubic and Quartic Reciprocity laws a long time ago, and I knew these proofs, but I don't know they could be proved using $m$-th power reciprocity law. I didn't find this law in the book. Speaking about $m$-th power reciprocity law I mean this
$K$ is a number field containing a primitive $n$-th root of unity, and $\alpha, \beta \in \mathcal O_K$ are relatively prime to each other and to $n$. Then 
$$\biggl(\frac{\alpha}{\beta}\biggr)_{n} \biggl(\frac{\beta}{\alpha}\biggr)_{n}^{-1}=\prod_{\mathfrak p \mid n\infty}\biggl(\frac{\alpha,\beta}{\mathfrak p}\biggr)_{n},$$
where $\biggl(\frac{\alpha,\beta}{\mathfrak p}\biggr)_{n}$ is the $n$-th power Hilbert symbol. Bibliography seems to be quite usefull, i'm going to check it, thanks for it.
 A: I like the book of Ireland and Rosen "A classical Introduction to Modern Number Theory".
There the Cubic and Biquadratic Reciprocity law are proved (and the Eisenstein Reciprocity law , the $m$-th power reciprocity law). Furthermore the book has a quite remarkable bibliography. 
A: A complete exposition of the derivation of Hilbert's product formula from Artin's reciprocity law, plus an application to cubic and quartic reciprocity, is indeed contained in the second volume of Hasse's report on class field theory. 
If you take Artin's reciprocity law for granted, the whole thing is little more than an exercise: by Artin, the Jacobi symbol $(\mu/\alpha)_\ell$ only depends on the residue class of $\alpha$ modulo the conductor of the extension 
$K(\sqrt[\ell]{\mu})$ of the field $K$ of $\ell$-th roots of unity, and from 
this observation the whole reciprocity law for $\ell = 3$ and $\ell = 4$ follows.
A slightly different approach would be using Furtwängler's trick, which is explained in the correspondence between Artin and Hasse
(the pdf of the German version is free; an English translation will appear at the end of 2013 from Springer).
