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日時: 2023年6月30日(金) 15:30-16:30
Date: June 30, 2023 15:30-16:30
会場:東京大学大学院数理科学研究科 大講義室
Place: Lecture Room, Graduate School of Math. Sci. Bldg.
Guy Henniart 氏（Université Paris-Saclay）
Guy Henniart (Université Paris-Saclay)
Did you say p-adic? (English)
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!