I was recently looking at the wikipedia page on Eisenstein Reciprocity, which says it "extends Quadratic Reciprocity." However, though the two do seem to be related, I don't completely understand how Eisenstein Reciprocity implies Quadratic Reciprocity. Can someone give me an explanation about why Eisenstein reciprocity implies quadratic reciprocity, or possibly cubic and quartic reciprocity.
2026-03-26 03:11:33.1774494693
Quadratic Reciprocity as a consequence of Eisenstein Reciprocity
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This is Theorem $12.8$, Corollary $12.9$ etc. in the following text of Franz Lemmermayer, who is an expert for reciprocity laws. There are many articles explaining also various consequences of Eisenstein's and Artin's reciprocity law.