1 1.1 n 3 X n + Y n = Z n Fermat Fermat Diophantus 2 Bachet x 2 + y 2 = z 2 Fermat Wiles 4 Kummer 5 Dedekind 6 ζ n 1 n ζ n =

Transcription

1 Noether Dedekind

2 1 1.1 n 3 X n + Y n = Z n Fermat Fermat Diophantus 2 Bachet x 2 + y 2 = z 2 Fermat Wiles 4 Kummer 5 Dedekind 6 ζ n 1 n ζ n = cos 2π + i sin 2π Fermat n n x n + y n = z n = (x + y)(x + ζ n y)(x + ζny) 2 (x + ζn n 1 y) = z n Z[ζ n ] = {a 0 + a 1 ζ n + + a r ζ r n r 0, a 0,, a r Z} Z x + ζ k ny (k = 0, 1,, n 1) z Fermat n Z[ζ n ] Z[ 5] 6 Z[ 5] 6 = 2 3 = (1 + 5)(1 5) 1 Pierre de Fermat ( ) 2 Diophantus ( ) Diophantus 3 Claude-Gaspard Bachet de Mèziriac ( ) 1612 Problèmes plaisants et delectables qui se font par les nombres 4 Andrew Wiles (1953-) Oxford 5 Ernst Kummer ( ) Weierstrass Kronecker Berlin Richard Dedekind ( ) Dedekind Dirichlet 2

3 2, 3, 1 + 5, 1 5 Z[ 5] 6 2 Kummer 2 = α 2, 3 = βγ = αβ, 1 5 = αγ α, β, γ (complex ideal number) Kummer Fermat Kummer Dedekind (ideal) Fermat 1.2 N N = 2 x 2 Ny 2 = = 1, = 1, = 1 x 2 Ny 2 = 1 Pell 7 x 2 Ny 2 = (x + Ny)(x Ny) = 1 Z[ N] x + Ny Z[ N] Z[ 2] {±(1+ 2) n n Z} x 2 2y 2 = 1 Dirichlet 8 7 John Pell ( ) Diophantus Pell Pell 8 Gustav Peter Lejeune Dirichlet ( ) ( ) Fermat n = 5 n = 14 3

4 Pythagoras 9 Fermat Fermat x n + y n = z n n = 2 x 2 + y 2 = z 2 Pythagoras Pythagoras Pythagoras Fermat 1.3 p p p p 4 1 p = x 2 + y 2 x, y p 4 3 p = x 2 + y 2 x, y Gauss 10 p 4 1 x 2 1 (mod p) x p 4 3 x p = = (2 + i)(2 i) 13 = = (3 + 2i)(3 2i) Z[i] = {a + bi a, b Z} 4 3 Z[i] Z[i] = (2 + i) 2 (2 i) 2 = = (3 + 2i) 2 (3 2i) 2 = Pythagoras ( ) Platon Pythagoras Pythagoras 10 Carl Friedrich Gauss ( ) Gauss 1801 Disquisitiones Arithmeticae Gauss 4

5 Fermat 17 1 (class number) (algebraic number field) 2.2 K O K n > 0 α n c 1 α n 1 c n = 0 c 1,, c n Z K α O K K (ring of integers) O K K K = Q(ζ n ) O K = Z[ζ n ] = { n 1 k=0 a k ζ k n a k Z} ζ n n cos (2π/n) + i sin (2π/n) ζ k n ζ n k K O K 2.3 m 1 1 K = Q( m), Z[ m] = {a b m a, b Z} m 2, 3 mod 4 O K = [ 1 m ] Z = {a b 1 m } a, b Z m 1 mod

6 m 1 1, K = Q( m) K = Q + Q m α = x + y m K (x, y Q) α = x y m K α O K α + α = 2x, αα = x 2 my 2 Z α O K α n +c 1 α n 1 + +c n = 0 c 1, c n Z φ(a+b m) = a b m φ (α ) n +c 1 (α ) n 1 + +c n = 0 α O K α+α, αα O K α + α, αα O K Q O K Q = Z a/b O K Q, (a, b) = 1 11 (a/b) n + d 1 (a/b) n d n = 0 n > 1, d 1,, d n Z a n = b(d 1 a n 1 + bd 2 a n d n b n 1 ) (a, b) = 1 b = ±1 a/b Z O K Q = Z α + α, αα O K Q = Z α + α Z, αα Z d 1 = (α + α ), d 2 = αα α 2 + d 1 α + d 2 = 0 α O K x, y Q (1) m 2, 3 mod 4 2x, x 2 my 2 Z x, y Z (2) m 1 mod 4 2x, x 2 my 2 Z 2x, 2y, x y Z p a 0 a p 2.4 p a 0 a p ord p (t) a = p m u v, m Z u, v p m ord p (a) a p m < 0 (1) ord p (ab) = ord p (a) + ord p (b) (2) ord p (a + b) min{ord p (a), ord p (b)} (3) ord p (a) ord p (b) ord p (a + b) = min{ord p (a), ord p (b)} 11 (a, b) = gcd(a, b) 6

7 2.4 x, y Q 2x, x 2 my 2 Z 2y Z p x 2 my 2 Z ord p (x 2 my 2 ) 0 ord p (x 2 ) 0 ord p (m) + 2ord p (y) < 0 ord p (x 2 my 2 ) = ord p (m) + 2ord p (y) < 0 ord p (m)+2ord p (y) 0 ord p (m) 1 2ord p (y) 1 ord p (y) 0 x 2 my 2 Z ord 2 (x 2 my 2 ) 0 ord 2 (x) 1 ord 2 (x 2 ) = 2ord 2 (x) 2 ord 2 (m) + 2ord 2 (y) < 2 ord 2 (x 2 my 2 ) = ord 2 (m) + 2ord 2 (y) < 2 ord 2 (m) + 2ord 2 (y) 2 ord 2 (m) 1 2ord 2 (y) 3 ord 2 (y) 1 z z Z p ord p (z) 0 2y Z (1) m 2, 3 mod 4 2x, x 2 my 2 Z x, y Z x, y Z 2x, x 2 my 2 Z 2x, x 2 my 2 Z x, y / Z x, y Z Z x 2 my 2 / Z Z x = 2x + 1, y = 2y + 1 (x, y Z) 2 2 x 2 my 2 = α + 1 m 4 / Z (α Z) (2) m 1 mod 4 2x, x 2 my 2 Z 2x, 2y, x y Z 2x, 2y, x y Z 2x, x 2 my 2 Z x, y Z x = 2x + 1, y = 2y + 1 (x, y Z) x, y Z 2 2 x = 2x + 1, y = 2y x 2 my 2 = α + 1 m 4 (α Z) 2x, x 2 my 2 Z 2x, x 2 my 2 Z x x = 2x + 1 x = x 2 (x Z) y 2y Z y = 2y + 1 y = y 2 (y Z) x y / Z x = 2x + 1 2, y = y x = x, y = 2y x 2 my 2 = α + 1 4, x2 + my 2 = β + m 4 7 (α, β Z) x 2 my 2 / Z

8 m 2, 3 mod 4 x + y m O K x, y Z O K = {a + b m a, b Z} m 1 mod 4 x + y m O K 2x, 2y, x y Z 2x, 2y, x y Z x, y Z x = 2x + 1, y = 2y (x, y Z) x + y m x y + 2y 1 + m 2, x y + (2y + 1) 1 + m α = x + y m O K 2 α = a + b 1 + m (a, b Z) O K = {a + b 1 + m } 2 2 a, b Z 2.3 K O K Z- O K Noether K O K Z-Noether K K a O K (fractional ideal) (1), (2) (1) O K 0 c ca O K 0 (2) a K 0 O K - (1), (2) O K Noether 3.2 K α αo K (α) (α) n 3.3 K K a, b a i b i (n 1, a i a, b i b) a b ab ab O K a (O K : a) = {x K xa O K } (O K : a) a (O K : a) = O K (O K : a) a 8 i=1

9 3.4 K (1) K O K K Cl(K) Cl(O K ) (class number) (2) K O K O K K (i), (ii), (iii) (i) Cl(K) (ii) O K (iii) O K (i) (ii) I O K P O K Cl(K) = I/P Cl(K) = I/P = {ē} I = P O K I = P ( ) O K (ii) (iii) (iii) (ii) O K p O K (0) p 0 α O K α = q 1 q 2 q n (q i : O K ) p q i p i (q i ) p Dedekind (0) (q i ) = p O K (ii) (i) 9

10 3.6 J O K 0 a, b J a b def a b O K - J/ Cl(K) ( ) f J I I/P = Cl(K) a, b J, f(a) = f(b) a b ( ) ( ) f(a) = f(b) a = (u)b u K g g : b a ; t ut g O K - a b ( ) a b O K - φ : a b 0 a, b a φ(ab) = aφ(b) = bφ(a) φ(a) a = φ(b) b ρ = φ(b)/b a a φ(a) = aρ φ ρa = b f(a) = f(b) ( ) a I a ba J b O K \{0} f(ba) = ba = a f f ( ) ( ) (ii) (i) 0 O K - (a) 0 φ a O K (a) φ a (x) = xa 10

11 O K - 0 O K J/ Cl(K) J/ = Cl(K) J/ = 1 Cl(K) = Z K = Q Cl(Q) = {ē} Q Z = {±1} a K a (a) g : K I Im g = P Coker g = I/ Im g = I/P = Cl(K) Ker g = O K 1 O K K g I Cl(K) 0 f f Ker f = Cokerf = Z[ 2] 7 7 = (3 + 2)(3 2) = ( )(5 3 2) = ( )( ) = (1) = (5 3 2)(1 + 2) 2 7 (1) Z[ 2] = {±(1 + 2) n n Z} K O K Z

12 4.1 ( ) Dirichlet 4.2 K (1) K K R (2) K K C σ σ(k) R σ σ : K C ( x σ(x)) 4.3 (Dirichlet ) K r 1 r 2 r = r 1 +r 2 1 O K = Z r ( ) K K = Q( 2) 2.3 O K = Z[ 2] 2 x 2 2 x 2 2 = (x + 2)(x 2) r 1 = 2, r 2 = 0 Dirichlet r = r 1 + r 2 1 = = 1 O K = Z Z/2Z O K = {±(1 + 2) n n Z} (Dirichlet Z Z/2Z ) O K = Z[ 2] = {a + b 2 a, b Z} Z[ 2] = {±(1 + 2) n n Z} (1 + 2)( 2 1) = Z[ 2] n ±(1+ 2) n Z[ 2] {±(1 + 2) n n Z} Z[ 2] 12

13 α Z[ 2] 1 Z[ 2] 1 α > 0 0 < β α < (1 + 2)β β {(1+ 2) n n Z} β Z[ 2] 1 α β < α β Z[ 2] α β 1 1 < α β < α β = x + y 2 (x, y Z) (x + y 2)(a + b 2) = 1 a b Z (ax + 2by) + (bx + ay) 2 = Q ax + 2by = 1 bx + ay = 0 bx ay = 0 (x y 2)(a b 2) = (ax + 2by) + ( bx ay) 2 = = 1 (x + y 2)(x y 2)(a + b 2)(a b 2) = (x 2 2y 2 )(a 2 2b 2 ) = 1 x 2 2y 2 = (x + y 2)(x y 2) = ±1 x y ±1 2 = x + y 2 α β = x + y 2 > 1 1 < x y ±1 2 = x + y 2 < 1 1 < x + y 2 < < 2x < x = 1 1 < x + y 2 < x = 1 1 < 1 + y 2 < < y 2 < 2 y Z α β = 1 α = β {±(1 + 2) n n Z} 5 ( ) a 5.1 p a p p {±1} F p a ( ) a x 2 a (mod p) x = 1 p ( ) a p = a, b Z p ( ) ( ) ( ) ab a b = p p p F p {a2 a F p } 13

14 Gauss 5.3 (1) q p ( ) ( ) q p = ( 1) p 1 2 q 1 2 p q (2) p 12 ( ) { 1 1 p 1 (mod 4) = ( 1) p 1 2 = p 1 p 3 (mod 4) K 2 R Q 2 K = Q( m) m 1 m < 0 { m m 1 (mod 4) N = 4m m 2, 3 (mod 4) 5.4 Z/NZ (Z/NZ) 0 C χ : (Z/NZ) C (mod N )Dirichlet Dirichlet χ m p ( ) m χ(p mod N) = p (#) ( (Z/NZ) N ) χ N a ( ( )) a θ(a) ( ) l l l m θ(a) 12 Euler 14

15 (1) m 1 (mod 4) θ(a) = 1 (2) m 3 (mod 4) a 1 (mod 4) θ(a) = 1 a 3 (mod 4) θ(a) = 1 (3) m a 1, 1 m (mod 8) θ(a) = 1 θ(a) = 1 ( ) a ( ) a θ(a) (Z/NZ) l C Dirichlet ( ) Dirichlet (#) (1) ( ) (#) m 1 (mod 4) ( ) m = ( ) p ( ) p l l m = l 1 l 2 l s l i ( ) ( ) ( ) ( ) m l1 l 2 l s = = ( 1) p 1 2 (1+ l p p ls ) p p l 1 l 2 l s = m l 1 l 2 l s 3 (mod 4) l 1 l 2 l s l i 3 (mod 4) i l j 1 (mod 4) l j 1 (mod 4) l j 1 2 l i 3 (mod 4) l i 1 2 l 1 1 2,..., l s 1 2 l l s l l s (+) (2), (3) l 1 l s 5.5 N χ (mod N )Dirichlet L(s, χ) = n=1 χ(n) n s (χ )Dirichlet L χ(n) n N χ(n mod N) n N

16 5.6 ( 2 ) K 2 m, N χ (#) Dirichlet h K K w K K 1 h K = w K 2 L(0, χ) = w K N L(1, χ) 2π 5.7 w K K = Q( 1) 4 K = Q( 3) 6 K 2 2 K = Q( m) (m Z, m < 0) α K N(α) α N(α) = 1 α K 1 = = O K 1 w K N(α) = 1 α K α 1 α O K Case 1. m 2, 3 (mod 4) 2.3 O K = Z[ m] α = a + b m O K 1 (a, b Z) n = m > 0 N(α) = a 2 b 2 m = a 2 + b 2 n (1) m = 1 N(α) = a 2 + b 2 = 1 α = ±1, ±i w K = 4 (2) m < 1 b 0 1 < b 2 n a 2 + b 2 n = N(α) α 1 b = 0 α = ±1 w K = 2 [ 1 + m ] Case 2. m = 1 (mod 4) 2.3 O K = Z α = 2 a + b 1+ m O 2 K 1 (a, b Z) n = m > 0 N(α) = (a b)2 + ( b 2 )2 n 16

17 (1) m = 3 N(α) = (a b) b2 = 1 b = 0, ±1 α ±1, ± 1 + 3, ± w K = 6 (2) m < 3 m 1 (mod 4) m 7 b 0 N(α) = (a b)2 + n 4 b2 n 4 b2 > 1 α 1 b = 0 N(α) = a 2 α ±1 w K = N χ : (Z/NZ) C (#) Dirichlet L(0, χ) = 1 N N aχ(a) a= K, m, N χ (#) Dirichlet h K = w N K aχ(a) 2N Q( 1) Q( 3) 5.9 K = Q( 1) w K = 4, m = 1 3 (mod 4) N = 4m = 4 h K = a=1 a=1 aχ(a) = 1 (1 χ(1) + 3 χ(3)) 2 ( ) ( ) 1 2 χ(1) = 1, χ(3) = = = h K = 1 ( ( 1)) =

18 K = Q( 3) w k = 6, m = 3 1 (mod 4) N = 3 h k = 6 3 aχ(a) = (1χ(1) + 2χ(2)) 2 3 a=1 ( ) a ( ) χ(a (mod 3)) = χ(1) = 1, χ(2) = 3 ( ) 2 = 1 h k = 1 3 Q( 1) Q( 3) 5.6 K = Q( 1) h K = w K N L(1, χ) = 4 L(1, χ) 2π π L(1, χ) = n=1 13 h K = 1 K = Q( 3) h K = 6 3 2π χ(n) n = = π 4 L(1, χ) = 3 3 π L(1, χ) L(1, χ) = π 3 3 Q( 3) 1 L(1, χ) π 3 3 h K = 3 3 L(1, χ) π h K = 1 L(1, χ) = < 1 h K = L(1, χ) < π π < 2 h K h K = 1 2 h K = w K N L(1, χ) L(1, χ) 2π h K 13 arctan(x) 18

19 1 2 Q( 1), Q( 2), Q( 3), Q( 7), Q( 11), Q( 19), Q( 43), Q( 67), Q( 163) Baker Stark A ( ) (1) R Abel (2) a, b, c A (ab)c = a(bc) (3) a, b, c A a(b + c) = ab + ac (a + b)c = ac + bc (4) a A ax = xa = a x A (5) a, b A ab = ba (4),(5) (1) (5) (1) 6.1.1(4) x ( ) x 1 A (2) 0 A (3) A 1 = 0 a A a = a1 = a0 = 0 A = {0}

20 6.1.3 A B B A (1) a, b B = a + b, a, ab B (2) 1 B A (1) a R ax = xa = 1 x A a A (2) A 0 A (3) A B a B = a 1 B B A a A a a 1 A A A A A M Abel (M + ) M A- A µ : A M M a, b A, x, y M (µ(a, x) ax ) (1) (a + b)x = ax + bx (2) a(x + y) = ax + ay (3) a(bx) = (ab)x (4) 1x = x A- M N a A x N ax N N M A A A A A A- A I, J, a, b, 20

21 A a (1) 1 a = a = A (2) a A = a = A (1) a A a = a1 a A a (2) x a A x 1 A x 1 x = 1 a (1) a = A A A A {0} A A M A-(M i ) i I M A- a A A- (1) i I M i (2) i I M i = { x i x i M i } (3) am = { a i x i a i a, x i M} (3) M A b ab = { ab a a, b b} ab A A a 1, a 2,, a n a 1 a 2 a n M A-N M A- M Abel M/N x M/N, a A ax = ax 21

22 well-defined A M/N M/N A- a A x, y A/a x y = xy A/a well-defined A/a well-defined x = x x x = α α a ax = ax = a(α + x ) = aα + ax = ax = ax x = x, y = y x x = α, y y = β α, β a x y = xy = (α + x )(β + y ) = αβ + αy + βx + x y = x y = x y A- M/N M N A/a A a A, B ϕ : A B (1) a, b A ϕ(a + b) = ϕ(a) + ϕ(b) (2) a, b A ϕ(ab) = ϕ(a)ϕ(b) (3) ϕ(1) = ϕ : A B (1) ϕ(0) = 0 (2) ϕ( a) = ϕ(a) (3) ϕ(a 1 ) = ϕ(a) 1 ( a A ) (1) ϕ(0) = ϕ(0 + 0) = ϕ(0) + ϕ(0) ϕ(0) = ϕ(0) ϕ(0) = 0 (2) (1) 0 = ϕ(0) = ϕ(a a) = ϕ(a) + ϕ( a) ϕ( a) = ϕ(a) 22

23 (3) 1 = ϕ(1) = ϕ(aa 1 ) = ϕ(a)ϕ(a 1 ) ϕ(a 1 ) = ϕ(a) M, M A- ϕ : M M A- A- (1) x, y M ϕ(x + y) = ϕ(x) + ϕ(y) (2) x M, a A ϕ(ax) = aϕ(x) A- ϕ : M N (1) ϕ(0) = 0 (2) ϕ( a) = ϕ(a) ι : A A A ι f : A B, g : B C g f A a A M A-N M A- ϕ : A a a A/a ψ : M x x M/N ϕ ψ A- A/a M/N ϕ : A B Ker ϕ = {a A ϕ(a) = 0} Im ϕ = {ϕ(a) B a A} Ker ϕ A Im ϕ B 23

24 x, y Ker ϕ a A ϕ(x y) = ϕ(x) ϕ(y) = 0 0 = 0 ϕ(ax) = ϕ(a)ϕ(x) = ϕ(a)0 = 0 x y, ax Ker ϕ Ker ϕ A α, β Im ϕ ϕ(x) = α, ϕ(y) = β x, y A α β = ϕ(x) ϕ(y) = ϕ(x y) Im ϕ αβ = ϕ(x)ϕ(y) = ϕ(xy) Im ϕ ϕ(1) = 1 Im ϕ Im ϕ B Ker ϕ ϕ Im ϕ ϕ ϕ : M N A- ϕ Ker ϕ, Im ϕ Ker ϕ = {x M ϕ(x) = 0} Im ϕ = {ϕ(x) N x M} Ker ϕ M A-Im ϕ N A- Ker ϕ Im ϕ α, β Im ϕ a A ϕ(x) = α, ϕ(y) = β x, y A α β = ϕ(x) ϕ(y) = ϕ(x y) Im ϕ aα = aϕ(x) = ϕ(ax) Im ϕ f : A B f Ker f = {0} ϕ : M N A- ϕ Ker ϕ = {0} 24

25 ( ) x Ker f f(x) = 0 f(0) = 0 f(x) = f(0) f x = 0 ( ) f(x) = f(y) f f(x y) = 0 x y Ker f x y = 0 f ϕ : M N A- L N A- ϕ 1 (L) M A- x, y ϕ 1 (L) ϕ(x), ϕ(y) L ϕ(x y) = ϕ(x) ϕ(y) L x y ϕ 1 (L) a A ϕ(ax) = aϕ(x) L ax ϕ 1 (L) ϕ : A B b B ϕ 1 (b) A α A a A ϕ(αa) = ϕ(α)ϕ(α) b αa ϕ 1 (b) ϕ : A B a A ϕ ϕ(a) B α, β ϕ(a) ϕ(x) = α, ϕ(y) = β x, y a x y a α β = ϕ(x) ϕ(y) = ϕ(x y) ϕ(a) b B ϕ(a) = b a A bα = ϕ(a)ϕ(x) = ϕ(ax) ϕ(a) A a A X = {b b A a b} Y = {c c A/a } X Y M A-N M A- S = {L L M A N L} T = {K K M/N A } S T 25

26 π : A A/a F : X Y G : Y X F (b) = π(b) G(c) = π 1 (c) b b = F (b) F (b ) c c = G(c) G(c ) G F = id X, F G = id Y f : A B a a Ker f A π : A A/a g π = f g : A/a B g a = Ker f ϕ : M N A- L L Ker ϕ M A-π : M M/L ψ π = ϕ A- ψ : M/L N ψ L = Ker ϕ π g π = f g A/a a = a a a a Ker f f(a) = f(a ) g : A/a B g(a) = f(a) g π = f a Ker g a Ker f 26

27 g a Ker g, a = 0 a Ker f, a a Ker f a g a = Ker f A x A 0 y A s.t. xy = 0 x A x n = 0 n N x A A N A A A x A A xy = 0 y A \ {0} x 1 A y = 1y = x 1 xy = 0 y A 0 x, y A xy = 0 = x = 0 y = 0 A A p( A) A xy p = x p y p A m( A) A A a m a = a = A a = m 27

28 A SpecA A p A A/p m A A/m p A x, y A xy p x p y p x, y A/p xy = 0 x = 0 y = 0 A/p m A m A A/m {0}, A/m ( ) A/m ( ) F B f : F B f f(1) = 1 Ker f F F Ker f = 0 f A a a m A m A Noether Noether ( 6.4.1) A Noether Zorn Σ a A a Σ Σ I = {a i i I} Σ i I a i 28

29 A x, y a i I x, y a j j I a A x y, ax a j i I a i i I 1 a i 1 a i a i A a i Σ a i I Σ Zorn Σ a A A A A A \ A A m x x ( ) m A \ A m m A m A \ A A \ A = m A \ A A A A \ A A \ A A f : A B p B f 1 (p) A 1 p 1 f 1 (p) f 1 (p) A xy f 1 (p) f(x)f(y) = f(xy) p f(x) p f(y) p x f 1 (p) y f 1 (p) a 1, a 2,, a n A p A n a i p i=1 a i p i p = a i p = a i i 29

30 i a i p i x i a i x i p x i Πx i Πa i a i p x i Πx i p a i p p = a i j p a j M A-L M AL = { a i x i a i A, x i L} AL M A- A L L {x 1, x 2,, x n } AL = Ax 1 + Ax Ax n L A Ax 1, x 2,, x n A (x 1, x 2,, x n ) x (x) PID (3) A a A- M am a (a) (a)m am (M i ) i I A- (M i ) i I M i = {(x i ) i I x i M i i x i = 0} i I A a A a(x i ) i I = (ax i ) i I M i A- (M i ) i I A ( ) A I A- A- I {1, 2,, n} A I A n 30

31 A M M A-M A- M = x 1 M + + x n M ϕ : A n M ϕ((a 1,, a n )) = a 1 x a n x n ϕ A- M A n / Ker ϕ A- ϕ : A n M e i = (0,, 0, 1, 0,, 0))( i 1 0) A n = e 1 A+ + e n A M = ϕ(a n ) = ϕ(e 1 )A + + ϕ(e n )A M A-a A M A- ϕ ϕ(m) am ϕ n + a 1 ϕ n a n = 0 n N, a i a M = Ax 1 + Ax Ax n x i ϕ(x i ) am ϕ(x i ) = Σ n j=1a ij x j a ij a ϕ(x i ) n a ij x j = 0 j=1 A A[t] M α M (Σa i t i )α = Σa i ϕ i (α) well-difined A[t] M n (δ ij t a ij )x j = 0 j=1 P = (δ ij t a ij ), x = t (x 1, x 2,, x n ) P x = 0 P P (c) (A[t] ) P (c) P x = (detp )Ix = (detp )x = 0 i (detp )x i = 0 x 1, x 2,, x n M detp A[t] x M (detp )x = 0 31

32 P (i, j)- δ ij t a ij detp t n monic n 1 a M M A- ϕ n + a 1 ϕ n a n = A N A A N A A N = p x N A x n = 0 n N p SpecA p x n p p x n p x n 1 p x p x N x N A A Σ Σ = {a a A n N, x n a} (0) Σ Σ Σ Zorn a Σ a, b A a, b a a a + (a) a a + (b) a a + (a), a + (b) Σ m, n N s.t. x m a + (a), x n a + (b) x m+n a + (ab) a + (ab) Σ a Σ ab a a a b a = ab a a Σ x a a x N 32

33 A a r(a) = {x A n N s.t. x n a} r(a) A x, y r(a) x n, y m a n, m N l = max{n, m} (x y) 2l a x y r(a) a A (ax) n ax r(a) r(a) a A a r(a) a A A/a A a 1 1 x r(a) n N s.t. x n a A/a n N s.t. x n = 0 x N A/a r(a) N A/a N A/a = P P SpecA/a A/a a A 1 1 r(a) = π 1 ( P) = π 1 (P) = p P SpecA/a P SpecA/a ( π : A A/a ) p A r(p) = p a p SpecA p r(p) x r(p) x n p n N p x n p x n 1 p x n 2 p x p r(p) p 33

34 a, b A r(a b) = r(a) r(b) x r(a b) n N s.t. x n a b n N s.t. x n a x n b x r(a) x r(b) x r(a) r(b) a, b A r(ab) = r(a) r(b) r(a + b) = r(r(a) + r(b)) x r(ab) n N s.t. x n ab n N, a i a, b i b s.t. x n = Σa i b i n N s.t. x n a b x r(a b) x r(a) r(b) x r(a + b) n N s.t. x n a + b n N, a a, b b s.t. x n = a + b n N s.t. x n r(a) + r(b) x r(r(a) + r(b)) a, b A r(a) + r(b) = A = a + b = A 34

35 r(a + b) = r(r(a) + r(b)) = r(a) = A 1 a + b a + b = A A a, b a + b = A a b A a 1, a 2,, a n n n a i = i=1 n n = 2 ab a b x a b a + b = 1 a a, b b i=1 a i x = x(a + b) = xa + xb ab n 3 Π n 1 i=1 a 1 = n 1 i=1 a 1 b = Π n 1 i=1 a 1 = n 1 i=1 a 1 1 i n 1 a i + a n = A x i + y i = 1 x i a i y i a n n 1 n 1 x i = (1 y i ) i=1 i=1 Π n 1 i=1 x i+y = 1 Y a n Π n 1 i=1 x i b b + a n = A n = 2 n a i = ba n = b a n = i=1 n i=1 a i A a, b a b (a : b) (a : b) = {x A xb a} b (x) (a : b) (a : x) 35

36 (a : b) A x, y (a : b) xb, yb a a A, b b (x y)b = xb yb a axb a x y, ax (a : b) A- M Ann(M) = {a A am = 0} Ann(M) A M ( annihilator) Ann(M) = 0 M A f : A B A a a f a e f(a) B a e = f(a)b = { x i y i x i f(a), y i B} B b b f ( )b c b c = f 1 (b) ( b c A ) f : A B a A b B (1) a a ec (2) b b ce (3) r(a) e r(a e ) (1), (2) (3) x r(a) e x = Σ m i=1y i b i b i B, y i f(r(a)) y i r(a) a i y i = f(a i ) i a n i i a n i N y n i i = f(a n i i ) f(a) a e N N x N a e ( n = max{n 1, n 2,, n m } N = mn ) (1), (2) 36

37 f : A B a A b B a ece = a e, b cec = b c A q( A) A xy q = x q n N s.t. y n q q A/q 0 A/q q A r(q) q A xy r(q) m N s.t. x m y m q = m N s.t. x m q n N s.t. y mn q x r(q) y r(q) r(q) A r(q) q A f : A B q B f 1 (q) A f π : B B/q π f f 1 (q) = f 1 (Ker π) π f = g ϕ g : A/f 1 (q) B/q ( ϕ : A A/f 1 (q) ) A/f 1 (q) B/q q A p p = r(q) q p A a q 1, q 2,, q n a = n i=1 a = q i a a 37 q i

38 A a a a = n i=1q i a = n i=1q i (1) i j = r(q i ) r(q j ) (2) i = 1, 2,, n j i q j q i (1) p- p-q 1, q 2 p- r(q 1 ) = r(q 2 ) = p r(q 1 q 2 ) = r(q 1 ) r(q 2 ) = p p = p xy q 1 q 2 y q 1 q 2 i (i = 1 i = 2) xy q i y q i x n q i n N x r(q i ) = p = r(q 1 q 2 ) x m q 1 q 2 m N q 1 q 2 p- (1) a A a = n i=1q i A B C A B C = A B q j q i {q 1, q 2,, q n } q i a j i a = q 1 q i 1 q i+1 q n (2) x A q A p- (1) x q (q : x) = A (2) x q (q : x) p- (3) x p (q : x) = q 38

39 (1), (3) (2) y (q : x) xy q x q y n q n N y r(q) = p q (q : x) p r((q : x)) = p (q : x) p-yz (q : x) y r(q) = p xyz q xz q z (q : x) a A a = n i=1q i a ( q i p i - ) {p 1, p 2,, p n } x A (a : x) = ( n i=1q i : x) = n i=1(q i : x) (1),(2) r((a : x)) = n i=1r((q i : x)) = x qj p i r((a : x)) r((a : x)) = p j j r((a : x)) p 1, p 2,, p n a = n i=1q i i x i q i x i j i q i (2) r((a : x i )) = p i {p 1, p 2,, p n } r((a : x)) a p i a {p 1, p 2,, p n } A S A A (1) 1 S (2) a, b S = ab S A a A 39

40 (1) A A \ {0} (2) A \ a a A (1) A a, b A, ab = 0 = a = 0 b = 0 a, b A \ {0}, ab 0 A \ {0} (2) A \ a a, b A, a, b a = ab a a, b A, ab a = a a b a a A A S A S (a, s) (b, t) def u S (at bs)u = 0 A S as as = 0 (a, s) (a, s) (a, s) (b, t) (at bs)u = 0 u S (bs at)u = 0 (b, t) (a, s) (a, s) (b, t) (b, t) (c, u) (at bs)v = (bu ct)w = 0 v, w S (at bs)uvw = atuvw bsuvw = 0 (bu ct)svw = bsuvw cstvw = 0 atuvw cstvw = (au cs)tvw = 0 (a, s) (c, u) S 1 A (a, s) S 1 A a/s S 1 A (a/s) + (b/t) = (at + bs)/st (a/s)(b/t) = ab/st well-defined S 1 A 40

41 6.2.6(1) well-defined a/s = a /s, b/t = b /t (as a s)u = 0, (bt b t)v = 0 u, v S as u = a su bt v = b tv (at + bs)s t uv = (as u)tt v + (bt v)ss u = (a su)tt v + (b tv)ss u = (a t + b s )stuv (at + bs)/st = (a t + b s )/s t (as u)(bt v) = (a su)(b tv) (abs t a b st)uv = 0 ab/st = a b /s t S 1 A A S A S 1 A a A, s, s S (1) s a/ss = a/s (2) s/s (3) 0/s (1) s(s a) (ss )a = 0 (2) (3) a/s = 0 a = 0 at = 0 t S a/s = A S A f : A S 1 A f(a) = a/1 f 41

42 a, b A f(a + b) = (a + b)/1 = a/1 + b/1 = f(a) + f(b) f(ab) = ab/1 = (a/1)(b/1) = f(a)f(b) f(1) = 1/1 = 1 f f f Ker(f) = {a A s S s.t. sa = 0} A f a Ker(f) a/1 = 0 s S s.t. sa = (1) A S = A \ {0} S 1 A A S 1 A FracA A (2) A p S = A \ p S A S 1 A A p A p (2) (1) 0 a/s S 1 A a 0 a S s/a S 1 A a/s S 1 A f : A S 1 A A S 1 A a/s as 1 A F as 1 F S 1 A A- M A S 42

43 A S A M A- M S (x, s) (y, t) def u S s.t. u(sy tx) = 0 M S S 1 M (x, s) S 1 M x/s x/s, y/t S 1 M a/u S 1 A x/s + y/t = (tx + sy)/st (a/u)(x/s) = ax/su S 1 M S 1 A f : M N A- S 1 f : S 1 M S 1 N S 1 f(x/s) = f(x)/s S 1 f S 1 A- S 1 f well-defined x/s = x /s t(s x sx ) = 0 t S ts x = tsx ts f(x) = tsf(x ) S 1 f(x/s) = f(x)/s = ts f(x)/tss = tsf(x )/tss = f(x )/s = S 1 f(x /s ) S 1 f S 1 f S 1 A- x/s, y/t S 1 M a/u S 1 A S 1 f(x/s + y/t) = S 1 f(tx + sy/st) = f(tx + sy)/st = f(x)/s + f(y)/t = S 1 f(x) + S 1 f(y) S 1 f((a/u)(x/s)) = S 1 f(ax/su) = af(x)/su = (a/u)(f(x)/s) = (a/u)s 1 f(x/s) S 1 f S 1 A f : M N A-S 1 f : S 1 M S 1 N x/s Ker S 1 f f(x)/s = 0 uf(x) = 0 u S ux Ker f ux = 0 x/s = ux/us = 0 43

44 A a, b A S A a b S 1 a S 1 b S 1 A- S 1 a, S 1 b S 1 A A a f : A S 1 A a e = S 1 Af(a) S 1 A- S 1 a S 1 A f : A S 1 A S 1 A I I = S 1 a A a x/s I s/1 S 1 A x/1 I x I c x/s I ce I I ce I ce I ( ) I = I ce a I c a a ece = a e a cec = a c ( ) A a a = a ec S 1 A 1 1 a S 1 a S A a, b A S 1 a S 1 b = S 1 (a b) α S 1 a S 1 b a a, b b, s, t S α = a/s = b/t u S (ta sb)u = 0 tua = sub a b a/s = tua/tus S 1 (a b) S 1 a S 1 b S 1 (a b) a/s S 1 (a b) a a b a/s S 1 a S 1 b 44

45 S A a A r(s 1 a) = S 1 r(a) (3) r(a) e r(a e ) S 1 r(a) r(s 1 a) x/s r(s 1 a) (x/s) n S 1 a n u(tx n s n a) = 0 t, u S a a u n t n x n = u n t n 1 s n a a utx r(a) x/s = utx/uts S 1 r(a) f : A S 1 A q A p- (1) S p S 1 q = S 1 A (2) S p = S 1 q S 1 p- (S 1 q) c = q (1) x S p x p = r(q) x n q n S x n S x n /1 S 1 q S 1 A (2) q ec = q q q ec ( ) x q ec f(x) = x/1 q e = S 1 q x/1 S 1 q a/s x/1 = a/s (sx a)t = 0 s, t S tsx = at q S p = ts p x q q ec q (S 1 q) c = q 45

46 S 1 q S 1 p-(x/s)(y/t) S 1 q y/t S 1 q (uxy sta)v = 0 u, v S a q uvxy q u, v S uv p xy q y q q x n q n N (x/s) n = x n /s n S 1 q r(s 1 q) = S 1 r(q) = S 1 p S 1 q S 1 p A S 1 A S 1 A A S f : A S 1 A a A a = n i=1q i a ( q i p i -) S p 1, p 2,, p m p m+1, p m+2,, p n m m S 1 a = S 1 q i, (S 1 a) c = i=1 S 1 a (S 1 a) c i=1 q i S 1 a = S 1 ( n i=1q i ) = n i=1s 1 q i n i=1s 1 q i = m i=1s 1 q i S 1 q i S 1 p i - p 1, p 2,, p m S 1 p 1, S 1 p 2,, S 1 p m ( i j S 1 p i = S 1 p j ) j i S 1 q j S 1 q i ( 1 i, j m) i f j i q j q i (1 i, j m) j i q j q i (1 i, j n) S 1 a = m i=1s 1 q i 46

47 m m (S 1 a) c = (S 1 q i ) c = i=1 a = n i=1q i i=1 q i a A a a p a (q ) S = A \ p S A p p A p p S (S 1 a) c = q S p p a ( ) q a A B α B A monic α n + a 1 α n a n 1 α + a n = 0 n N a i A x A A B x B (1) x B A (2) A[x] A- (3) B A- C A[x] C (4) A- A[x]- M 47

48 (1)= (2) x n +a 1 x n 1 + +a n = 0 n N a 1, a 2,, a n A x n = (a 1 x n a n ) x n {x n 1, x n 2,, x, 1} A A[x] = Ax n 1 + Ax n A (2)= (3) C = A[x] (3)= (4) M = C C y A[x] yc = 0 y 1 = y = 0 M A[x]- (4)= (1) ϕ : M M ϕ(y) = xy a = A M A[x]- ϕ(m) = xm M ϕ ϕ n + a 1 ϕ n a n = 0 n a i A M x n + a 1 x n a n M x n + a 1 x n a n = x 1, x 2,, x n B A A[x 1, x 2,, x n ] A- n n = n 1 A n 1 = A[x 1, x 2,, x n 1 ] A- x n A n 1 A n = A[x 1, x 2,, x n ] = A n 1 [x n ] A n 1 - y 1, y 2,, y m A A n 1 z 1, z 2,, z l A n 1 A n A n {y i z j 1 i m, 1 j l} A A A B C A B x, y C A[x, y] A- A[x y], A[xy] A[x, y] 6.3.2(3) x y, xy A x y, xy C 48

49 6.3.5 A B B A B A B A B A A B B A B B A A B C B A C B x B C x n + c 1 x n c n = 0 n N c 1, c 2,, c n C C = A[c 1, c 2,, c n ] C A-x C C [x] C - C [x] A x A x C A B B A ι : A B q B A p p = ι 1 (q) = q A q B p A A/p B/q π : B B/q Ker π ι = p A/p B/q x B x n + a 1 x n a n = 0 n N, a i A B/q x x n + a 1 x n a n = 0 B/q A/p A/p 0 x B/q x n + a 1 x n a n = 0 n N a i A/p n a n = (x n + a 1 x n a n 1 x) = x(x n 1 + a 1 x n a n 1 ) 49

50 B/q a n 0 (a n = 0 x n 1 + a 1 x n a n 1 = 0 n ) x 1 = a n 1 (x n 1 + a 1 x n a n 1 ) a n 1 A/p B/q x 1 B/q B/q B/q 0 y A/p y 1 B/q y 1 A/p y m + a 1 y 1 m + + a m = 0 m N, a i A/p ym 1 y 1 = (a 1 + a 2 y + a m y m 1 ) A/p A/p A B B A q q B q A = q A = q = q p = q A = q A x B x n + a 1 x n a n = 0 n N, a i A x/s B p (x/s) n + (a 1 /s)(x/s) n 1 + (a 2 /s 2 )(x/s) n (a n /s n ) = 0 x/s A p B p A p m p A p m A n, n q, q B p n n n A p = n A p = m (n A p = q p A p = (q A) p = p p = m n ) n, n B p n n n = n q = q A B C B A S A S 1 C S 1 B S 1 A 50

51 x/s S 1 C x C, s S x C A n N a i A x n + a 1 x n a n = 0 (x/s) n + (a 1 /s)(x/s) n a n /s n = (x n + a 1 x n a n )/s n = 0 x/s S 1 A S 1 C S 1 A y/t S 1 B S 1 A (y/t) m + (b 1 /t 1 )(y/t) m b m /t m = 0 n N, b i A, t i S u = t 1 t 2 t m (tu) m ((yu) m + b 1 t 2 t 3 t m t(yu) m b m t 1 t 2 t m 1 t m u n 1 )v = 0 v S yuv A yuv C y/t = yuv/tuv S 1 C A B a A B x a x a monic a B B a A B C B A a e A a C B a r(a e ) x B a x n + a 1 x n a n = 0 n N, a 1, a 2,, a n a x A x C x n = (a 1 x n 1 + a 2 x n a n ) x n a e x r(a e ) 51

52 x r(a e ) n N x n a e x n = m a i x i i=1 m N, a i a, x i C x i A M = A[x 1, x 2,, x m ] A- x n M m a i x i M am i=1 M ϕ x n x n a x a A B x B A a x K = FracA K x r(a) x a K x x 1, x 2,..., x n K L x i x K x K[t] x i x x i a K x Π(t x i ) x 1, x 2,, x n a a K A K r(a e ) = r(a) r(a) L/K α L ϕ α : L L v L ϕ α (v) = αv ϕ α K- L K ϕ α A α T : L K v L T (v) = TrA α T L T L/K Tr L/K Tr L/K (v) v ψ : L L K ψ(x, y) = Tr L/K (xy) ψ K- L/K ψ 52

53 x L ψ(x, y) = 0 y = 0 y L ψ(x, y) = 0 x = A K = FracA L K B L A K L {v 1, v 2,, v n } B n Av i i= v L v K a 0 v r + a 1 v r a r = 0 r N, a i K A a i A a r 1 0 a 0 v A L K A B L K u 1, u 2,, u n L K u i B (i = 1, 2,, n) L/K ψ 0 x L ψ(x, ) : L K K- ψ(u i, v j ) = Tr L/K (u i v j ) = δ ij L K v 1, v 2,, v n x B x x = n x i v i (x i K) i=1 u i B xu i B y L y K t m + a 1 t m a m Tr L/K (y) = [L : K(y)]a (a = A ) Tr L/K (xu i ) A 53

54 n Tr L/K (xu i ) = Tr L/K ( x j u i v j ) = j=1 n Tr L/K (x j u i v j ) = j=1 n x j Tr L/K (u i v j ) = j=1 n x j δ ij = x i j=1 x i A x n i=1 Av i B n i=1 Av i 6.4 Noether Dedekind A (1) A I (2) A (3) A a 1 a 2 n a 1 a 2 a n = a n+1 = a n+2 = (1)= (2) A a I a A {0} I I I a 0 a a 0 x a \ a 0 a 1 = a 0 + (x) a 1 a 0 a 1 a 0 a = a 0 a (2)= (3) A (a i ) i N a = i N a i A a a = (x 1, x 2,, x r ) x 1, x 2,, x r a n n N a = a n a n = a n+1 = (3)= (1) A I a I a b b I I Noether (1) (3) 54

55 6.4.4 A Noether A S S 1 A Noether A a A/a Noether S 1 A A a a ec = a 1 1 S 1 A A/a A a 1 1 A/a A Noether A p A p Noether B ϕ : A B B Noether A p B ϕ B A A/a A Noether B B A- B Noether A- M M A- M Noether A- A, B B B A- B Noether A- B Noether Notether A M Noether M A A n A n Noether A- n A n Noether A- A π : M N A- N ker π Noether A- M Noether A- M A- M 1 M 2 π(m 1 ) π(m 2 ) (M 1 ker π) (M 2 ker π) 55

56 π(m n ) = π(m n+1 ) = (M n ker π) = (M n+1 ker π) = n M n = M n+1 = M Noether A a,b,c a = b c = a = b a = c a Noether A Σ = {a a A} Σ Σ a a a = b c A b, c( a) a b, c a Noether A a A xy a x a n N y n a (a : (y)) (a : (y 2 )) (a : (y 3 )) (a : (y n )) = (a : (y n+1 )) n N a (a + (x)) (a + (y n )) a = α + sx = β + ty n α, β a, s, t A ay = yα + sxy a ay = yβ + ty n+1 ty n+1 = 56

57 ay yβ a t (a : (y n+1 )) = (a : (y n )) ty n a a = β + ty n a (a + (x)) (a + (y n )) a a (a + (x)) (a + (y n )) a = (a + (x)) (a + (y n )) x, y n a a + (x), a + (y n ) a a n N y n a a A Noether (1) A Noether a A r(a) n a n N A Noether r(a) r(a) = (x 1, x 2,, x m ) x i r(a) i x k i i a k i N n N ( n = Σ m i=1(k i 1) + 1 ) Σl i = n l 1, l 2,, l m N x l 1 1 x l 2 2 x l m a r(a) n = A{x l 1 1 x l 2 2 x l m Σl i = n} a A a 0 a 1 a n n A A dima Noether A 0 A 0 a A A m A a = m n n N A 1 m A a m- A Noether r(a) n = m n a n N n m n a m n 1 a 57

58 m 0 a m r((a)) = m m n (a) m n 1 (a) n N m n 1 \(a) b m n 1 \ (a) x = a/b FracA x 1 A x 1 = b/a A = b = b/1 = (b/a)(a/1) = x 1 a (a) b x 1 A A x 1 A x 1 m m (x 1 m m m A[x 1 ]-FracA m A[x 1 ]- A Noether m A- x 1 A ) x 1 = b/a y m by m n (a) by = az z A x 1 y = by/a = az/a = z A x 1 m A x 1 m A m x 1 m = A m = Ax = (x) (x 1 m = A x m ) m n a x n a A m = (x) c A k N cx k a \ m n ( a x n ) a \ m n dx l ( d A, l < n) x l a (x n 1 ) = m n 1 a n a \ m n = a = (x n ) = m n Noether Dedekind Dedekind 0 0 q Dedekind A q = m n m n N Dedekind 1 0 q( 0) r(q) 58

59 q m- m A m A m Dedekind q S 1 q S 1 q A m S 1 m S 1 q = (S 1 m) n = S 1 m n m n m m n m- q = m n A Dedekind 0 a A a A m 1, m 2,...,m s ( ) a = m 1 m 2 m s ( ) a A Noether a a = n i=1q i a ( q i n i -) A 1 A 0 n 1, n 2,..., n n q 1, q 2,...,q n n i=1q i = Π n i=1q i a = Π n i=1q i a q i Π n i=1q i q i n i=1q i = Π n i=1q i n i = r(q i ) n i a q i a q i a a = n i=1q i = Π n i=1q i a K O K Dedekind K n Q K/Q n O K Zv i K Q {v 1, v 2,, v n } O K Z- Z PID Noether i=1 59

60 O K Noether O K O K 1 O K 0 p O K (0) O K (0) Z = (0) p Z (0) p Z Z p O K [1],,, I(,2005) [2] M.F.Atiyah,I.G.MacDonald, Introduction to Commutative Algebra (Westview Press,1969) [3],, (,2011) [4] Paulo Ribenboim,, (,1983) [5], (,1986) [6], 2 (,2010) 60