数学まんじゅう - 暇つぶし2ch

数学まんじゅう at MATH

1:１３２人目の素数さん
23/12/13 20:57:20.46 b2RK67pK.net
おいしいいいいいいい

2:１３２人目の素数さん
23/12/14 07:23:53.64 8x5j3sRc.net
x^2 + y^2 + z^2
= (x + y + z)^2 - 2(xy + yz + zx)

3:１３２人目の素数さん
23/12/14 07:44:34.98 8x5j3sRc.net
x^3 + y^3 + z^3
= (x + y + z)^3 - 3(x + y + z)(xy + yz + zx) + 3xyz

(t - x)(t - y)(t - z)
= t^3 - (x + y + z)t^2 + (xy + yz + zx)t - xyz
(x + y + z)^3
= x^3 + y^3 + z^3 + 3x^2 y + 3x^2 z + 3x y^2 + 3x z^2 + 3y^2 z + 3y z^2 + 6xyz

4:１３２人目の素数さん
23/12/14 08:03:53.14 8x5j3sRc.net
P_n := (x_1)^n + ... + (x_m)^n
f(T) := (T - x_1) ... (T - x_m)
=: T^m - S_1 T^(m-1) + ... + (-1)^m S_m
S_1 = x_1 + ... + x_m
S_2 = x_1 x_2 + x_1 x_2 + ... + x_(m-1) x_m
...
S_m = x_1 ... x_m
f(x_i) = 0 (0 ≤ i ≤ m)
∴ -P_m = - S_1 P_(m-1) + ... + (-1)^m S_m

5:１３２人目の素数さん
23/12/14 08:36:03.98 8x5j3sRc.net
k[x, y]/(x + y, xy)
x^2 + y^2 = (x + y)^2 - 2xy
x^2 = (x + y)^2 - y(x + y) - xy
y^2 = (x + y)^2 - x(x + y) - xy

f: k[x] → k[x, y] → k[x, y]/(x + y, xy)
[x] = f(x)
[y] = f(-x +(x + y))
∴f: 全射
x^2∈Ker(f)
g∈Ker(f)
g = q x^2 + r (deg(r) < 2)
f(g) = [r] = 0
r = 0
∴ Ker(f) = (x^2)
∴ k[x, y]/(x + y, xy) ～ k[x]/(x^2)