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529: １３２人目の素数さん [] 2023/02/17(金) 16:32:18.19 ID:PDN8ps3Q

>>526 追加
>https://en.wikipedia.org/wiki/Quadratic_reciprocity
>Quadratic reciprocity

＜関連追加引用＞
History and alternative statements
The theorem was formulated in many ways before its modern form: Euler and Legendre did not have Gauss's congruence notation, nor did Gauss have the Legendre symbol.
In this article p and q always refer to distinct positive odd primes, and x and y to unspecified integers.

There is no kind of reciprocity in the Hilbert reciprocity law; its name simply indicates the historical source of the result in quadratic reciprocity. Unlike quadratic reciprocity, which requires sign conditions (namely positivity of the primes involved) and a special treatment of the prime 2, the Hilbert reciprocity law treats all absolute values of the rationals on an equal footing. Therefore, it is a more natural way of expressing quadratic reciprocity with a view towards generalization: the Hilbert reciprocity law extends with very few changes to all global fields and this extension can rightly be considered a generalization of quadratic reciprocity to all global fields.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1615510393/529

530: １３２人目の素数さん [] 2023/02/17(金) 16:33:02.23 ID:PDN8ps3Q

>>529
つづき

Connection with cyclotomic fields
The early proofs of quadratic reciprocity are relatively unilluminating. The situation changed when Gauss used Gauss sums to show that quadratic fields are subfields of cyclotomic fields, and implicitly deduced quadratic reciprocity from a reciprocity theorem for cyclotomic fields. His proof was cast in modern form by later algebraic number theorists. This proof served as a template for class field theory, which can be viewed as a vast generalization of quadratic reciprocity.

Robert Langlands formulated the Langlands program, which gives a conjectural vast generalization of class field theory. He wrote:[27]

I confess that, as a student unaware of the history of the subject and unaware of the connection with cyclotomy, I did not find the law or its so-called elementary proofs appealing. I suppose, although I would not have (and could not have) expressed myself in this way that I saw it as little more than a mathematical curiosity, fit more for amateurs than for the attention of the serious mathematician that I then hoped to become. It was only in Hermann Weyl's book on the algebraic theory of numbers[28] that I appreciated it as anything more.
(引用終り)
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http://rio2016.5ch.net/test/read.cgi/math/1615510393/530

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