16 : 2015/11/4(23:14) (1891) (1995) (2002) 7 8 IT 20 ( ),,, 1979,,, 2010,,, , 13,, Évariste Galois

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1 (1891) (1995) (2002) 7 8 IT 20 ( ),,, 1979,,, 2010,,, , 13,, Évariste Galois Evgraf Stepanovich Fedorov ) Andrew John Wiles, Pierre de Fermat n 3, n N x n + y n = z n 0 (x, y, z) 6 Grigory Yakovlevich Perelman, Jules-Henri Poincaré S 3 1

2 ,,, 2012,,, 2013,,, 2014, 14,, I II 2 I II B.C.3000?-B.C,1000? 10 Alkwarizmi,780?-850? 11 Niels Henrik Abel Gerolamo Cardano Niccolò Fontana 1499?-1557 [Tartaglia] 15 Ludovico Ferrari,

3 5 x 5 6x + 3 = 0 3 x 5 = 2 x = x 5 + x 4 4x 3 + 3x 2 3x + 1 = a( 0), b, c Q 2 ax 2 + bx + c = 0 x = b ± b 2 4ac 2a 2 (1) a, b, c b 2 4ac (2) b 2 4ac b2 4ac (3) b b2 4ac (4) b ± b 2 4ac 2a

4 G x, y G x y G G 1 i 1 + 2i 2 + i II I 2,,, 2,,, I 26 pp I I 26 pp25 4

5 I a + b 2 a + b 2 Q 2 F α F α F (α) 5 3 Q ( 3 ) a + b 3 a + b 3 Q (1) f 2 x, y K, x y f (x) f (y) (2) f y x y K, x, y = f (x) (3) x, y K f (x y) = f (x) f (y) (3) x, y K f (x + y) = f (x) + f (y) x, y K f (x y) = f (x) f (y) x, y K f (x y) = f (x) f (y) x, y K f (x y) = f (x) f (y) 5

6 ( ) 6 Q 2 ( (1) p, q Q f p + q ) 2 = p + q 2 ( (2) p, q Q f p + q ) 2 = p q 2 II p + q 2 p q 2 p + qi p qi ( ) 7 Q 2 f (1) f (0) = 0 (2) f (1) = 1 (3) n N, f (n) = n (4) m Z, f (m) = m (5) q Q, f (q) = q ( ) Q 2 p p (1) x x = 0 f (x x) = f (0) f (x) f (x) = f (0) f (0) = 0 (2) x 0 x 1 = x f (x 1) = f (x) f (x) f (1) = f (x) f (x) (f (1) 1) = 0 f (x) 0 f (1) = 1 (3) f (n) = f ( ) = f (1) + f (1) + + f (1) = = n (4) m = n f (m) = f ( n) = f (n) n = m (5) m Z, n N q = m n f (q) = f ( m n ) = f (m) f (n) = m n = q 8 x 2 = 2 x = ± ( ) 2 Q ! = 6 4 4! = 24 f ( x 2) = f (2) {f (x)} 2 = 2 f (x) = ± ( ) 2 f 2 = ± 2 ( p, q Q f p + q ) ( ) 2 = f (p)+f (q) f 2 = p±q 2 6

7 3.4 II 9 a, b Q 2 x 2 + ax + b = 0 α, β α + β = a α 2 + aα + b = 0 2 a 2 x 2 + ax + b = x 2 + ax + b 0 = x 2 + ax + b ( α 2 + aα + b ) = x 2 α 2 + ax aα + b b = (x α) (x + α) + a (x α) = (x α) (x + a + α) β = a α α + β = a 10 a, b Q 2 x 2 + ax + b = 0 2 α β α = β a α β 11 2 α Q α Q Q (α) 2 α Q α Q Q (α) p 1, q 1, p 2, q 2 Q p 1 + q 1 α, p 2 + q 2 α α 2 = aα b (p + qα) (a pq ) + α 7

8 α 2 + aα + b = 0 α 2 = aα b p + qα (p + qα) (a pq ) + α = pa p2 q + qaα + qα2 = q ( α 2 + aα ) + pa p2 q = qb + pq p2 q a, b Q 2 x 2 + ax + b = 0 α 2 Q (α) β β = a α Q (α) ( ) 2 Q (α) Q 2 2 f (x) Q (α) x f : Q (α) Q (α) x Q, f (x) = x f : Q (α) Q (α) f (0) = Q (α) f : Q (α) Q (α) 2 p + qα Q (α) f f (p + qα) f (p + qα) = f (p) + f (q) f (α) = p + qf (α) Q (α) f α f f (α) α 2 + aα + b = 0 f f ( α 2 + aα + b ) = f (0) f (α) 2 + af (α) + b = 0 8

9 2 f (α) 2 + af (α) + b = 0 2 x 2 + ax + b = 0 f (α) = α f (α) = β f (p + qα) = p + qα f (p + qα) = p + qβ f p, q Q, α, β 2 x 2 + ax + b = 0 f (α) = β, f (β) = α x Q (α), f (f (x)) = x α + β = a 2 f (α + β) = f ( a) f (α) + f (β) = a f (α) = β f (β) = a f (α) = a β = a 15 Q (α) f f (x) = x x p, q Q, α, β 2 x 2 + ax + b = 0 f f (p + qα) = p + qβ p + qα = p + qβ q (α β) = 0 α + β = a α = β α = β = a α 2 q = 0 x = p x 9

10 f f ( (α β) 2) (α β) 2 α β α β α + β = a α f ( f (α β) 2) = f ((α β) (α β)) = f (α β) f (α β) = (f (α) f (β)) (f (α) f (β)) = (β α) (β α) = (α β) 2 (α β) 2 Q D = (α β) 2 α β = ± D α + β = a α (α β) 2 (α β) 2 17 II 2 ax 2 + bx + c = 0 α + β = b a, α β = c a α β α β α β 2 (α β) 2 (α β) 2 α + β α β x 2 3x 1 = 0 α, β (1) α 2 + β 2 β (2) α + α (3) (α β) 2 β II 26 pp46 10

11 Q (α) α β α + β = a 3 3 {(α β) (β γ) (γ α)} ( ) Artin,E., (),,, , 30,, ,,, ,,, ,,, vol39-4, pp38-58, April, (1796 )Carolus Fridericus Gauss, p = 2 2e 1 p 11

17 ( :52) α ω 53 (2015 ) 2 α ω 55 (2017 ) 2 1) ) ) 2 2 4) (α β) A ) 6) A (5) 1)

17 ( :52) α ω 53 (2015 ) 2 α ω 55 (2017 ) 2 1) ) ) 2 2 4) (α β) A ) 6) A (5) 1) 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