3 3 1 α ω 53 (2015 ) 2 α ω 55 (2017 ) 2 1) 2000 2) 5 2 3 4 2 3 5 3) 2 2 4) (α β) 2 3 4 5 20 A 12 20 5 5 5) 6) 5 20 12 5 A (5) 1) Évariste Galois(1811-1832) 2) Joseph-Louis Lagrange(1736-1813) 18 3),Niels Henrik Abel(1802-1829) 4) 1
3 3 4 5 2 2 2 Q (α) (α β) 2 α + β II II 2 ax 2 + bx + c = 0 2 α, β α + β = b, αβ = c a a 1 2 2x 2 4x + 3 = 0 2 α, β (1) (α + 2) (β + 2) (2) β α + α β (3) (α β) 2 (4) α 3 β 3 2 7) 1(3)(α β) 2 8) 5) Felix Christian Klein(1849-1925) 6) Plato(B.C.427-B.C.347) 5 7) Albert Girard(1595-1632) 8) α β 2 2
1 1 1 2 n n 2 α, β α + β, αβ 2 3 α, β, γ 1 x 1 n n a 1, a 2, a 3,, a n 1, a n n x n 1 x n 2 x n 3 k=1 n 1 i,j n a k n a i a j 1 i,j,k n a i a j a k n x n p ( 1) p a i1 a i2 a ip 1 i 1,i 2,,i p n ( 1) n a 1 a 2 a 3 a n 1 a n n n n n 9) 10) 11) 12) 13) 19 14) 9) Gottfried Wilhelm Leibniz(1646-1716) 10) Leonhard Euler(1707-1783)18 11) Carolus Fridericus Gauss(1777-1855) 12) Georg Ferdinand Ludwig Philipp Cantor(1845-1918) 13) Julius Wilhelm Richard Dedekind(1831-1916) 14) https://www.ms.u-tokyo.ac.jp/activity/documents/tsuji.pdf 3
2 C n a n x n +a n 1 x n 1 +a n 2 x n 2 + +a 2 x 2 +a 1 x+a 0 = 0 C n a 0, a 1, a 2,, a n C a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 = a n (x r 1 ) (x r 2 ) (x r n ) r 0, r 1, r 2,, r n C 3 3 ax 3 + bx 2 + cx + d = 0 α, β, γ x, y, z 4 x, y (1) xy 2 z 3 x 2 yz 3 (2) xyz (3) xy + yz + zx (4) x 2 y + y 2 z + z 2 x 5 x yy zz x (1) xy 2 z 3 x 3 yz 2 (2) xyz (3) xy + yz + zx (4) x 2 y + y 2 z + z 2 x x yy zz x 15) 4(3) 5(4) x, y x 2 y + y 2 z + z 2 x xy 2 + yz 2 + zx 2 x yy zz x x 2 y + y 2 z + z 2 x x 2 y + y 2 z + z 2 x x, y, z x, y, z 6 3! = 6 4(1) 5(1) xy 2 z 3 x 2 y + y 2 z + z 2 x 6 2 β α 2 (β α) 2 15) C(3) 4
16) 3 D (3) 3 x, y, z 6 x, y, z xy 2 + z xy 2 + z xy 2 + z 120 x y y z z x x y z x xy 2 + z ( ) 240 x z y x z y x z y x xy 2 + z ( ) x x y z z y y z xy 2 + z ( ) 16) D (3) 5
y y x z z x x z xy 2 + z ( ) z z x y y x x y xy 2 + z ( ) 7 x 2 y + y 2 z + z 2 x 6 x 2 y + y 2 z + z 2 x ( ) 120 x 2 y + y 2 z + z 2 x ( ) 240 x 2 y + y 2 z + z 2 x ( ) x x 2 y + y 2 z + z 2 x ( ) y x 2 y + y 2 z + z 2 x ( ) z x 2 y + y 2 z + z 2 x ( ) x 2 y + y 2 z + z 2 x x, y, z 120 240 xy 2 + yz 2 + zx 2 x 2 y + y 2 z + z 2 x 17) 3 f x 1, x 2,, x n f x 1, x 2,, x n s 1, s 2,, s n x 1, x 2,, x n f s 1, s 2,, s n 17) http://www.kurims.kyoto-u.ac.jp/ kenkyubu/kokai-koza/h16-mukai.pdf 6
n a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 = 0 n n (1) a n x n +a n 1 x n 1 +a n 2 x n 2 + +a 2 x 2 +a 1 x+a 0 = 0 x 1, x 2,, x n (2) x 1, x 2,, x n 2 8 2 ax 2 + bx + c = 0 α, β α, β (1) α + β = (2) αβ = (3) (α + β) 2 = (4) α 2 + β 2 = (5) (α β) 2 = (6) α β = (7) α = (8) β = α β α β β α 1 2 1 1 (α β) 2 = α 2 2αβ + β 2 2 a, b, c 3 4 3 ( α + ωβ + ω 2 γ ) ( α + ωβ + ω 2 γ 9 (1) ( α + ωβ + ω 2 γ ) (2) ( α + ωβ + ω 2 γ ( α + ωβ + ω 2 γ ) 6 ( α + ωβ + ω 2 γ ( α + ωβ + ω 2 γ )3 = ( α 3 + β 3 + γ 3) + 3ω ( α 2 β + β 2 γ + γ 2 α ) + 3ω 2 ( αβ 2 + βγ 2 + γα 2) + 6αβγ ( α + ωβ + ω 2 γ αβγ 7 α 2 β + β 2 γ + γ 2 α αβ 2 + βγ 2 + γα 2 7
18) ( α + ωβ + ω 2 γ ) 3 2 α β 3 3 2 x 1, x 1, x n x i, x j i < j x i x j 2 2 3 2 x 1, x 2 x 1 x 2 3 x 1, x 2, x 3 (x 1 x 2 ) (x 1 x 3 ) (x 2 x 3 ) 10 11 4 x 1, x 2, x 3, x 4 3 (x 1 x 2 ) (x 1 x 3 ) (x 2 x 3 ) x 1 x 2 (x 1 x 2 ) x 1 x 2 (x 2 x 1 ) ( 1 k (x 1 x k ) (x 2 x k ) (x 2 x k ) (x 1 x k ) (x 1 x 2 ) ( 1) ( 1) ( 1) 3 ( 1) 12 6 (1) (2) 120 (3) 240 (4) x (5) y (6) z 18) D (3) C (3) 8
x 1 x 2 x 1 x 2 1 1 x 1, x 2,, x n n n 2 α β n ( 1) 2 3 4 5 13 3 x 3 + ax 2 + bx + c = 0 α, β, γ a, b, c (1) α + β + γ = (2) αβ + βγ + γα = (3) αβγ = 3 14 3 x 3 + ax 2 + bx + c = 0 α, β, γ a, b, c (1) α 2 + β 2 + γ 2 = (2) α 2 β 2 + β 2 γ 2 + γ 2 α 2 = (3) αβ 2 + βγ 2 + γα 2 + α 2 β + β 2 γ + γ 2 α = 3 2 a, b, c 15 3 (α β) (α γ) (β γ) a, b, c (1) (α β) (β γ) β a, b (2) (β γ) (γ α) γ a, b (3) (γ α) (α β) α a, b (4) 2 {(α β) (α γ) (β γ)} 2 {(α β) (α γ) (β γ)} 2 = {(α β) (β γ) (γ α)} 2 (i) α 2 β 2 γ 2 = (ii) αβγ (αβ + βγ + γα) = (iii) αβγ (α + β + γ) = 9
(iv) αβ 2 + βγ 2 + γα 2 + α 2 β + β 2 γ + γ 2 α = (v) α 2 β 2 + β 2 γ 2 + γ 2 α 2 = (vi) α 2 + β 2 + γ 2 = (vii) α + β + γ = (viii) αβ + βγ + γα = (ix) αβγ = (5) (α β) (α γ) (β γ) a, b, c 16 ω 1 1 3 ω 2 + ω + 1 = 0 3 x 3 + ax 2 + bx + c = 0 3 α, β, γ D = (α β) (α γ) (β γ) (i) D = (α β) (α γ) (β γ) (ii) ( α + ωβ + ω 2 γ a, b, c, D (iii) ( α + ω 2 β + ωγ a, b, c, D (iv) (α + β + γ) + ( α + ωβ + ω 2 γ ) + ( α + ω 2 β + ωγ ) = 3α 3 (v) a = 0, b = p, c = q (3) 3 19) 4 3 20) 120 240 21) 4 8 22) 23) 5 5 24) 12 20 25) 5 2000 19 x 3 + px + q = 0 1 3 ω x = 3 q ( q 2 + 3 + q ( q 2 3 2 + 2 3, ω 3 q ( q 2 + + ω 2 3 q ( q 2 3 2 + 2 3, ω 2 3 q ( q 2 + 3 + ω q ( q 2 3 2 + 2 3 20) D (3) 21) C (3) 22) 4 S (4) 23) 4 A (4) 24) 5 S (5) 25) 5 A (5) 10
5 2000 α ω http://www.shimagami.net/aw/aw56.pdf [1] Coxeter,H,S,M, (), (),, 2009. [2] Sautor,M, (),,, 2007. [3], 30,, 1989. [4] Klein,F., (), 20 5,, 2012. [5], 14,, 2013. [6],,, 1989. [7],,, 2012. [8],, 2011vol39-4,pp38-58,, 2011. [9],,, 2010. [10], 13,, 2011. [11],,, 2016. [12],, 2 α ω 53, pp47-57, 2015. [13], α ω 55, pp.54-65, 2017. 11