2 1 1 α = a + bi(a, b R) α (conjugate) α = a bi α (absolute value) α = a 2 + b 2 α (norm) N(α) = a 2 + b 2 = αα = α 2 α (spure) (trace) 1 1. a R aα =

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1 1 1 α = a + bi(a, b R) α (conjugate) α = a bi α (absolute value) α = a + b α (norm) N(α) = a + b = αα = α α (spure) (trace) 1 1. a R aα = aα. α = α 3. α + β = α + β 4. αβ = αβ 5. β 0 6. α = α ( ) α = α β β 7. α β α + β α + β 8. αβ = α β 9. α β = α (β 0) β 10. N(α) = N(α) 11. N(αβ) ( ) = N(α)N(β) α 1. N = N(α) (β 0) β N(β) 13. S(α + β) = S(α) + S(β) [] S(α) = α + α = a α = a + bi a, b Z[i] Z[i] = {α α = a + bi, a, b Z} Z[i] i 1/16

2 3 3 α, β α = βκ + ρ, κ, ρ N(ρ) N(β) (1) [ ] ξ = α β = x + yi κ = [ x + 1 ] [ + y + 1 ] i () ξ κ 1 α β κ 1 α βκ β ρ = α βκ (1) N(α βκ) N(β) [ ] (1) () 1 κ, ρ i = (7 i)κ + ρ i (13 + 5i)(7 + i) = = 7 i 53 [ 81 κ = ] [ i 53 ] i = + i ρ = i (7 i)( + i) = i (16 + 3i) = 3 + i i = (7 i)( + i) + ( 3 + i) κ = 1 + i /16

3 4 ρ = i (7 i)(1 + i) = i (9 + 4i) = 4 N(ρ) = 16 < i + i i 1 + i + i ρ = 0 α = βκ α β β α α β β α ( ) ρ 0 α βκ α β β α Z 4 0 α, β (α, β) α, β {α, β} [α, β] 3 α 1, α, α 3,, α n β β α 1 γ 1 + α + γ + α 3 γ α n γ n 3/16

4 4 α 1 γ 1 + α + γ + α 3 γ α n γ n β = α 1 β γ 1 + α β + γ + α 3 β γ α n β γ n [ ] 4 α, β, γ, λ µ µ λ µ = κλ + ρ, N(ρ) N(λ) ρ = µ κλ 3 ρ α, β, γ, 0 λ N(ρ) N(λ) ρ = 0 µ = κλ α, β, γ, [ ] 5 α, β, γ, µ δ µ δ λ α µ δ α µ, δ α λ β, γ, λ λ α, β, γ, N(λ) N(µ) λ µ 0 N(λ) N(µ) λ = µ δ µ [ ] 5 (unit,einheit) 6 Z[i] ±1, ±i 4 4/16

5 4 [ ] 1 ɛ = a + bi 1 ɛ = ɛ ɛ ɛ ɛɛ = 1 N(ɛɛ ) = N(ɛ)N(ɛ ) = 1 N(ɛ) = 1 a + b = 1 (a, b) = (±1, 0), (0, ±1) ɛ = ±1, ±i ɛ = ±1, ±i ɛ ±1, ±i 4 [ ] 6 α α β (associate) α α, α, iα, iα β 4 7 α, β λ µ αβ = ɛλµ ɛ λ α, β λ = αβ = βα (3) αβ α, β αβ λ αβ = δλ (4) (3) (4) αβ = δαβ = δβα α = δα, β = δβ δ α, β µ = δκ (5) α, β µ δα = α δκ, δβ = β δκ 5/16

6 5 (6) (3) α = α κ, β = β κ (6) λ = αβ κ = βα κ λ κ = αβ = βα λ ( ) λ κ α, β N(κ) > 1 N < N(λ) λ κ κ = ɛ (5) µ = δɛ δ = ɛµ (7) (7) (4) αβ = ɛλµ [ ] α, β α, β (α, β) = 1 8 (α, β) = 1 α βγ α γ (α, β) = 1 α, β αβ βγ α βγ α, β βγ αβ βγ αβ = γ α γ α [ ] 5 7 α α 0 α α 0 9 α N(α) α [ ] α β, γ α = βγ N(α) = N(β)N(γ) 6/16

7 5 β, γ α [ ] Z[i] Z[i] [ ] β, γ = βγ N() = N(β)N(γ) N(β)N(γ) = 4 N(β) = N(γ) = β = 1 ± i, 1 ± i, γ = 1 ± i, 1 ± i (1 + i)(1 i) [ ] i [ ] 1 + i β, γ 1 + i = βγ N(1 + i) = N(β)N(γ) N(β)N(γ) = N(β) = 1 or N(γ) = 1 β, γ 1 + i [ ] 4 3,5,7,11,13 [ ] 3 β, γ 3 = βγ N(3) = N(β)N(γ) β, γ 3 N(β)N(γ) = 9 N(β) = N(γ) = 3 7/16

8 5 5 β, γ 5 = βγ N(β)N(γ) = 5 N(β) = N(γ) = 5 β = 1 + i, γ = 1 i 5 7 β, γ 7 = βγ N(β) = N(γ) = 7 β, γ 7 11 β, γ 11 = βγ N(β) = N(γ) = 11 β, γ β, γ 13 = βγ N(β) = N(γ) = 13 β = 3 + i, γ = 3 i m + 3 p = 4m + 3 β, γ p = βγ N(β) = N(γ) = p β = a + bi p a + b a, b a = k, b = l + 1 a + b = 4k + 4l + 4l + 1 4m + 3 a a, b p 11 ρ ρ αβ ρ α ρ β 8/16

9 5 (ρ, α) 1 α ρ ρ α (ρ, α) = 1 8 ρ β 1 4m + 1 p = 4m + 1 ( ) 1 = ( 1) p 1 = ( 1) m = 1 p a, b a + 1 = bp (a + i)(a i) = bp 11 p p a + i p a i p(c + di) = a + i p(c + di) = a i pd = ±1 p 13 4m + 1 ππ p = 4m + 1 p π 1, π p = π 1 π N(p) = p N(π 1 ) = N(π ) = p 9 π 1, π π 1 = x 1 + y 1 i, π = x + y i x 1 + y 1 = x + y = p (8) x 1 x y 1 y + (x 1 y + x y 1 )i = p x 1 x y 1 y = p (9) x 1 y + x y 1 = 0 (10) (8),(9) (x 1 x ) + (y 1 + y ) = 0 x 1 = x, y 1 = y π 1 = π 9/16

10 5 π 1, π π 1 = x 1 y 1 + x 1 y 1 i π x 1 + y 1 (8) x 1 = 0 y 1 = 0 x 1 = y 1 5 π 1 π 1 π π = aπ 1 π 1 = x 1 + y 1 i, π = x + y i π 1 π = x 1 x y 1 y + (x 1 y + x y 1 )i x 1 y = x y 1 (11) π 1 x 1, y 1 x = ax 1, y = by 1 (11) bx 1 y 1 = ax 1 y 1 a = b π = aπ π ππ p = ππ p π π p 1 p p = ab p > a > 1, p > b > 1 π π a π b p p 10/16

11 5 Z[i] π i. 4m N(π) 4m /16

12 θ θ θ N 1 < N < N 3 < θ θ α 1 α 1 θ = α 1 α α θ θ θ = α 1 α α 3 = β 1 β β 3 β 4 α 1, α, α 3, β 1, β, β 3, β 4 α 1 θ β 1 β β 3 β 4 α 1 β 1 θ α 1 = β 1 α α 3 = β β 3 β 4 β 4 = 1 θ = α 1 α α 3 = β 1 β β 3 α 1 = β 1, α = β, α 3 = β i i i i i 1. N( + 9i) = = 85 = 5 17 (1 + i)(4 + i) = + 9i 1/16

13 6. N(5 + 7i) = = 74 = 37 (1 + i)(6 + i) = 5 + 7i 3. N(11 10i) = 1 = i 11 10i + 3i = 8 53i 11 10i i, = = 4 + i 13 3i i = ( 3i)(4 + i) i = ( i) = (1 + i)(1 i)( i) 1 + i N( i) = = 185 = i 1 + i = i 5 = 1 + 6i + 16i = (1 + i)(1 i)(1 + i)(1 + 6i) 5. N( i) = = 4810 = i 1 + i i 1 + i i + 3i = = = i 75 80i i 13 = i = i = 6 i i = (1 + i)(1 + i)( + 3i)(6 i) 16 p p 4m + 1 p p = a + b p a, b a = l + 1, b = n 13/16

14 7 p = 4(l + l + n ) + 1 p 4m + 1 p 4m , π p = ππ π = a + bi p = a + b 7 p 4m + 1 ( ) 1 = 1 p x (mod p) x(1 < x < p) x (p x) + 1 = x px + p (mod p) x (mod p), 1 < x < p x + 1 = kp, 1 < x < p k < p x + y = kp (1) k u x (mod k), k < u k v y (mod k), k < v k u, v u + v 0 (mod k) 14/16

15 7 u + v = tk, (t k ) (13) (1) (13) (x + y )(u + v ) = pk t (xv yu) + (xu + yv) = pk t (14) u x, v y (mod k) xv yu xy xy 0 (mod k) xu + yv x + y 0 (mod k) (14) k kz = xv yu, kw = xu + yv z + w = pt, t k z, x, t x, y, k (1) t = 1 a + b = p 7 1. p = 13. p = ( ) 1 = 1 13 x (mod 13) x = 13 u = 1, v = = 4, = = = 13 15/16

16 = 97 5 u =, v = 1 ( ) + (44 + 1) = = = m + 1 (a + b )(c + d ) = (ad bc) + (ac + bd) (15) 18 n 1 x, y n = x + y n α = x + yi n = (x + yi)(x yi) = αα α = π 1 π π 3 π k π 1, π, π 3,, π k p p α, p x, p y (x, y) = 1 π 1, π, π 3,, π k n = N(α) = N(π 1 )N(π )N(π 3 ) N(π k ) = p 1 p p 3 p k p 1, p, p 3,, p k 4m + 1 n (15) n [1] 1997 [] /16

ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4

ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4 20 20.0 ( ) 8 y = ax 2 + bx + c 443 ax 2 + bx + c = 0 20.1 20.1.1 n 8 (n ) a n x n + a n 1 x n 1 + + a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 444 ( a, b, c, d