Colloquium
Seminar information archive ~05/24|Next seminar|Future seminars 05/25~
Organizer(s) | AIDA Shigeki, OSHIMA Yoshiki, SHIHO Atsushi (chair), TAKADA Ryo |
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URL | https://www.ms.u-tokyo.ac.jp/seminar/colloquium_e/index_e.html |
2023/06/30
15:30-16:30 Room #大講義室(auditorium) (Graduate School of Math. Sci. Bldg.)
If you do not belong to Graduate School of Mathematical Sciences, the University of Tokyo, please apply from the form at https://forms.gle/z22nKn1NUrT41qiR7
Guy Henniart (Université Paris-Saclay)
Did you say p-adic? (English)
If you do not belong to Graduate School of Mathematical Sciences, the University of Tokyo, please apply from the form at https://forms.gle/z22nKn1NUrT41qiR7
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 pr where p runs through prime divisors of N and pr is the highest power of p dividing N. Now work modulo p, modulo p2, modulo p3, etc. You have invented the p-adic integers, which are, I claim, as real as the real numbers and (nearly) as useful!
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 pr where p runs through prime divisors of N and pr is the highest power of p dividing N. Now work modulo p, modulo p2, modulo p3, etc. You have invented the p-adic integers, which are, I claim, as real as the real numbers and (nearly) as useful!