2 1 1 2000 2 (1891) 17 2 3 (1995) 4 5 6 (2002) 7 8 IT 20 ( ),,, 1979,,, 2010,,, 2010 13, 13,, 2011 13 1 Évariste Galois 1811-1832 2 Evgraf Stepanovich Fedorov 1853-1919) 1885 20 3 Andrew John Wiles, 1953-4 Pierre de Fermat 1607-1665 5 n 3, n N x n + y n = z n 0 (x, y, z) 6 Grigory Yakovlevich Perelman, 1966-7 Jules-Henri Poincaré 1854-1912 8 3 3 S 3 1
,,, 2012,,, 2013,,, 2014, 14,, 2013 3 17 I II 2 I 2 9 10 II 2 2 2 2 2 2 2 5 2 3 4 2 3 5 11 2 3 12 13 14 4 15 4 2 3 9 B.C.3000?-B.C,1000? 10 Alkwarizmi,780?-850? 11 Niels Henrik Abel 1802-1829 12 Gerolamo Cardano 1501-1576 13 3 14 Niccolò Fontana 1499?-1557 [Tartaglia] 15 Ludovico Ferrari, 1522-1565 2
5 x 5 6x + 3 = 0 3 x 5 = 2 x = 5 2 2 5 5 x 5 + x 4 4x 3 + 3x 2 3x + 1 = 0 2.1 2 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 3 4 5 3 2 2 2 3 3 4 3.1 3
G x, y G x y G G 1 i 1 + 2i 2 + i II I 2,,, 2,,, I 26 pp25 2 16 0 I 3 2 16 I 26 pp25 4
3.2 4 2 + 3 2 1 + 2 2 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 3 3.3 (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 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 (1 + 1 + + 1) = f (1) + f (1) + + f (1) = 1 + 1 + + 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 2 2 2 2 3 3! = 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
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
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 3.5 2 2 2 12 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) = 0 13 2 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
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 2 3.6 14 2 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
4 2 16 2 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 18 2 2 2 α β α β 2 (α β) 2 (α β) 2 α + β α β 2 17 19 2 x 2 3x 1 = 0 α, β (1) α 2 + β 2 β (2) α + α (3) (α β) 2 β 17 2 18 19 II 26 pp46 10
5 2 3 2 2 Q (α) α β α + β = a 3 3 {(α β) (β γ) (γ α)} 2 5 2 2 2 3 5 ( ) 1 17 20 1. 8 2. Artin,E., (),,, 2010 3., 30,, 1996 4.,,, 1989 5.,,, 2012 6.,,, vol39-4, pp38-58, April, 2011 20 (1796 )Carolus Fridericus Gauss,1777-1855 p = 2 2e 1 p 11