Timeline for Is there a proof of quadratic reciprocity using $p$-adic numbers?
Current License: CC BY-SA 4.0
9 events
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| Apr 7, 2019 at 7:47 | comment | added | Seewoo Lee | but of the course the proof of the product formula must involve global arguments, so is global in nature (as any proof of the quadratic reciprocity law must be at least in the sense that the statement of this theorem involves two primes) - I strongly agree with this and that's why there might not exist purely local proof (at least what I want, which is ambiguous though). What I thought was 'can we find a relation between the existence of $\sqrt{p}$ in $\mathbb{Q}_{q}$ and $\sqrt{q}$ in $\mathbb{Q}_{p}$?', which seems very hard to deal with only two primes, not all the primes. | |
| Apr 5, 2019 at 23:03 | vote | accept | Seewoo Lee | ||
| Apr 5, 2019 at 23:03 | |||||
| Apr 4, 2019 at 12:45 | comment | added | Victor Protsak | The proof using Gauss's lemma is known as Gauss's third proof. | |
| Apr 4, 2019 at 12:06 | comment | added | Olivier | @FrançoisBrunault Thanks François and Keith, that's fascinating. | |
| Apr 4, 2019 at 9:13 | comment | added | KConrad | The excerpt from Milnor's book is available at people.reed.edu/~ormsbyk/kgroup/resources/GaussQuadratic.pdf. It has Tate's calculation of $K_2(\mathbf Q)$ on p. 101 and says on p. 102 "Tate remarks that his proof of this theorem is lifted directly from the argument which was used by Gauss in his first proof of the quadratic reciprocity law." | |
| Apr 4, 2019 at 8:57 | comment | added | François Brunault | Salut Olivier, I checked in the Disquisitiones and Gauss' proof uses induction on the primes $p,q$. Interestingly in n°139, Gauss introduces a symbol $[x,y]$ which is almost the Hilbert symbol! (more precisely $[x,y]$ is the product of the $(x,y)_p$ where $p$ runs through the prime factors of $y$). But I don't know whether this is Gauss' first proof. | |
| Apr 4, 2019 at 8:31 | comment | added | Olivier | Salut François "and is very similar to Gauss' first proof of quadratic reciprocity" really? Isn't Gauss's first proof though Gauss's lemma, so counting multiples of $a$ in intervals? Is that really very similar? | |
| Apr 4, 2019 at 8:09 | comment | added | François Brunault | One can find a nice proof of the product formula in Milnor "Introduction to algebraic K-theory", Chapter 11. According to Milnor, this proof of quadratic reciprocity is due to Tate and is very similar to Gauss' first proof of quadratic reciprocity. | |
| Apr 4, 2019 at 7:59 | history | answered | Olivier | CC BY-SA 4.0 |