Seminar information archive ~06/07Next seminarFuture seminars 06/08~

Organizer(s) ABE Noriyuki, IWAKI Kohei, KAWAZUMI Nariya (chair), KOIKE Yuta

Future seminars


15:30-16:30   Room #大講義室(auditorium) (Graduate School of Math. Sci. Bldg.)
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Guy Henniart (Université Paris-Saclay)
Did you say $p$-adic? (English)
[ Abstract ]
I am a Number Theorist and $p$ is a prime number. The $p$-adic numbers are obtained by pushing to the limit a simple idea. Suppose that you want to know which integers are sums of two squares. If an integer $x$ is odd, its square has the form $8k+1$; if $x$ is even, its square is a multiple of $4$. So the sum of two squares has the form $4k$, $4k+1$ or $4k+2$, never $4k+3$ ! More generally if a polynomial equation with integer coefficients has no integer solution if you work «modulo $N$» that is you neglect all multiples of an integer $N$, then a fortiori it has no integer solution. By the Chinese Remainder Theorem, working modulo $N$ is the same as working modulo $p^r$ where $p$ runs through prime divisors of $N$ and $p^r$ is the highest power of $p$ dividing $N$. Now work modulo $p$, modulo $p^2$, modulo $p^3$, etc. You have invented the $p$-adic integers, which are, I claim, as real as the real numbers and (nearly) as useful!