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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.

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  • $\begingroup$ @Igor: Eisenstein's -th power reciprocity law is the natural generalization of cubic and biquadratic reciprocity. It can be generalized further of course, e.g., by Hilbert's and Artin's reciprocity law. Artin's reciprocity law implies the other ones, see for example www.bprim.org/cyclotomicfieldbook/rlmain.pdf‎ $\endgroup$ Commented May 10, 2013 at 11:05
  • $\begingroup$ @Dietrich Burde: The link isn't working. "Page under construction. Will be updated soon!" $\endgroup$
    – Igor
    Commented May 10, 2013 at 14:30
  • $\begingroup$ @Dietrich Burde. Thx for this article, I found new solution of quadric reciprocity using the m-th power reciprocity law, but there isn't one for cubic and biquadratic. Also I proved the cubic reciprocity law via exercises in the book of Cox. And I'm still looking for proof of the biquadratic using m-th power reciprocity law. I'll be glad for any help or link. $\endgroup$
    – Igor
    Commented May 11, 2013 at 17:34

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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.

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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).

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