x 3 a (mod p) ( ). a, b, m Z a b m a b (mod m) a b m 2.2 (Z/mZ). a = {x x a (mod m)} a Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} a + b = a +

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1 1 22 1 x 3 (mod ) 2 2.1 ( )., b, m Z b m b (mod m) b m 2.2 (Z/mZ). = {x x (mod m)} Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} + b = + b, b = b Z/mZ 1 1 Z Q R Z/Z

2.3 ( ). m {x 0, x 1,..., x m 1 } modm 2.4 ( ). (, m) = 1 x b (mod m) modm 2.5. Z/Z 0 (mod ) x 1 (mod ) x 2.4 2.6. ( 1 + 2 + + k ) 1 + 2 + k (mod ). 1, 2,..., k n 1 1 n 2 2... n k k (n 1 + n 2 + + n k = )! n 1!n 2!... n k! 0 (mod ) 2.7 ( ). (, ) = 1 1 1 (mod ) 2.8 ( ). 1 1 g (Z/Z) 2 g 3 {1, 2..., 1} = {g, g 2,... g 1 = 1} 3 3.1 x 2 (mod m) x m m 2 Z/Z 0( ) 3 G g G x x = g n (n ) G g

3.2 3.1. ( ) { = 1 ( ) 1 ( ) ( ) = 0 4 3.2. ( ) () = ±1 1 1 ( 1)/2 ( ) (b) g l (mod ) (g ) = ( 1) l (c) ( ) ( ) b = ( ) b (d) Euler ( ) ( 1)/2 (mod ) ( ) ( ). () = 1 2, 2 2,... ( 1) 2 = 1 = 1 1 2, 2 2,... ( 1) 2 mod 1 2 ( 1) 2, 2 2 ( 2) 2,..., ( 1 2 )2 ( + 1 2 )2 (mod ) ( ) 2 b 2 (1 b ( 1)/2) = 1 1 2, 2 2,... ( 1 ( ) 2 )2 ( 1)/2 = 1 ( 1)/2 (b) (c) (b) ( ) 4 F ( 5.1)

(d) (b)(c) (c), q ( ) q 3.3 ( )., q ( ) ( ) ( ) q = ( 1) 1 2 q 1 2 q ( ) 1 = ( 1) 1 2 ( ) 2 = ( 1) 1 8 (2 1) 4 Z[ω] 4.1. () Z[ω] = { + bω, b Z, ω = e 2πi 3 } (b) ξ = + bω ξ = + bω 2 ξ (c) η ξ ζ Z[ω] ξ = ηζ (d) ϵ 1 ϵ Z[ω] ϵ

(e) ξ = + bω N(ξ) = ξξ = ( + bω)( + bω 2 ) = 2 b + b 2 ξ (f) η = ϵξ(ϵ ) η ξ η ξ (g) Z[ω] Z[ω] 5 4.2 ( ). () N(ξη) = N(ξ)N(η) (b) ϵ N(ϵ) = 1 (c) Z[ω] ±1, ±ω, ±ω 2 6 (d) N(ξ) ξ (e) 0 Z[ω] (f) γ, γ 1 0 κ Z[ω] γ = κγ 1 + γ 2 N(γ 2 ) < N(γ 1 ) (g) 6 (h) π π βγ π β π γ (i) Z[ω] Z[ω] (i) 1 ω (ii) q 2 (mod 3) q (iii) N(π) = ππ = ( ) π Z[ω] (iii) 1 (mod 3), q, π mod 3 1 mod3 2 Z[ω] 5 Z 6

(j) π Z[ω]/πZ[ω] N(π) { { + bω 0 < q, 0 b < q} (π = q ) {0, 1,... 1}(ππ = ) modπ (k) π (α, π) = 1 α N(π) 1 1 (mod π) (l) π. () (h) 7 (i) 1 ω 3 ξ q N(ξ) N(q) = q 2 N(ξ) = 2 b+b 2 4 2 4b+b 2 = (2 b) 2 0 1 (mod 3) N(ξ) = q ξ q q 1 (mod 3) 3.3 ( ) ( ) ( ) 3 1 3 ( = = ( 1) 1 2 ( 1) 3) 1 2 3 1 2 ( ( ) 1 = = = 1 3) 3 2 + 3 = ( + 3)( 3) = ( + 1 + 2ω)( 1 2ω) Z ( + 1 + 2ω) ( 1 2ω) Z ± 1 ± 2 ω = πγ(π, γ ) 2 = N(π)N(γ) N(π) = N(γ) = π = N(π) = ππ = ππ 7

(j) Z[ω] modπ S S modπ S modπ (k) t t = N(π 1) t (mod π)(t mod π ) t 0 (mod π) N(π) 1 1 (mod π) (l) Z[ω] Z Z[ω] 5 5.1 ( ). Z/Z F 8 F {0} F C {0} C χ : F C, b F χ(b) = χ()χ(b) χ ε F ε() = 1 χ ε χ(0) = 0 ε ε(0) = 1 F 5.2 ( ). () χ(1) = 1 (b) 0 (χ()) 1 = 1 (c) χ( 1 ) = χ() 1 = χ() 5.3. χ ε F t F χ(t) = 0 8 F finite field

. χ ε χ() 1 F S = χ()s = t F χ(t) = χ() 1 S = 0 t F χ(t) = S( F ) t F χ(t) 5.4 ( ). ˆF = {χ χ F } χλ() = χ()λ() χ 1 () = χ() 1 ˆF ε 5.5. ˆF 1. F g F = gl (l ) χ() = χ(g) l χ χ(g) χ(g) 1 1 1 λ(g) = ζ = e 2πi 1 λ 0 = ε, λ, λ 2,..., λ 2 ˆF ˆF λ 1 5.6. 1 χ() = 0 χ F ˆ. λ 5.5 χ χ() = S λs = χ = χ = χ λ()χ() λχ() χ() = S( ˆF ) λ() 1 S = 0

5.7 ( ). χ ˆF, F g (χ) = t F χ(t)ζ t g 1 (χ) = g(χ) ζ 1 e 2πi 5.8. 0, χ ε g (χ) = χ( 1 )g(χ). g (χ)χ() = χ() t χ(t)ζ t = t = t χ(t)ζ t χ(t)ζ t = g(χ) 5.9. χ ε g(χ) =, g(χ)g(χ) = χ( 1). g (χ)g (χ) F 0 g (χ) = χ()g(χ) g (χ) = χ()g(χ) = χ( 1 )g(χ) g (χ)g (χ) = g(χ)g(χ) = g(χ) 2 = 0 g 0 (χ) = t χ(t) = 0 F g (χ)g (χ) = ( 1) g(χ) 2 g (χ)g (χ) = x χ(x)ζ x y χ(y)ζ y = χ(x)χ(y)ζ x y x,y

g (χ)g (χ) = χ(x)χ(y)ζ x y x,y = χ(x)χ(y) x,y ζ (x y) ζ nt n = 0, n 0 0 t = x=y χ(x)χ(y) = x χ(x)χ(x) = ( 1) (χ(0) = 0 ) g(χ) 2 = χ( 1) = ±1 χ( 1) = χ( 1) g(χ) = t χ(t)ζ t = χ( 1) t χ( t)ζ t = χ( 1)g(χ) g(χ)g(χ) = g(χ)g(χ) = χ( 1) 5.10 ( ). χ, λ ˆF J(χ, λ) = χ()λ(b) +b=1,b F 5.11 ( ). χ, λ ε J(χ, χ 1 ) = χ( 1) χλ ε J(χ, λ) = g(χ)g(λ) g(χλ) J(χ, λ) =

. J(χ, χ 1 ) = +b=1 b 0 χ( b ) = χ( 1 ) 1 1 1 = 1 = 1 = c c 1 F {1} c F { 1} J(χ, χ 1 ) = c 1 χ(c) = c χ(c) χ( 1) = χ( 1) g(χ)g(λ) = ( x χ(x)ζ x )( y λ(y)ζ y ) = χ(x)λ(y)ζ x+y x,y = χ(x)λ(y)ζ t t x+y=t = ζ t χ(x)λ(y) t x+y=t t = 0 χ(x)λ(y) = χ(x)λ( x) x+y=0 x = λ( 1) χλ(x) x = 0 ( χλ ε) t 0 x = tx, y = ty χ(x)λ(y) = χλ(t) χ(x )λ(y ) x+y=t x +y =1 = χλ(t)j(χ, λ) g(χ)g(λ) = t χλ(t)ζ t J(χ, λ) = g(χλ)j(χ, λ)

J(χ, λ) = g(χ)g(λ) g(χλ) g(χ) = J(χ, λ) = 5.12. 1 (mod n), χ ˆF 9 n g(χ) n = χ( 1)J(χ, χ)j(χ, χ 2 )... J(χ, χ n 2 ) 5.13. χ 3 1 (mod 3) g(χ) 3 = J(χ, χ). 5.11 g(χ) 2 = J(χ, χ)g(χ 2 ) g(χ 2 ) = g(χ)( χ 3) g(χ) 2 = J(χ, χ)g(χ) g(χ) 3 = g(χ)g(χ)j(χ, χ) = χ( 1)J(χ, χ)( 5.9) = χ 3 ( 1)J(χ, χ) = J(χ, χ) 6 9 χ l = ε l

6.1 ( ) 2 ( ) t ζ t ( ) t = χ () g = g (χ ) = ( ) t ζ t, g = g 1 t χ () = ±1 χ = χ 5.9 g(χ )g(χ ) = g 2 (χ ) = χ ( 1) = ( 1) 1 2 ( 1) ( 1) 2 = ( ) g q 1 = (g 2 ) (q 1)/2 (mod q) ( 3.2(d)) q ( ) g q g (mod q) q g q = ( ( ) t ζ t ) q q t ( ) q t ζ tq (mod q) ( 2.6) q t q = t ( ) t ζ tq = g q = q ( q 1 ) g ( ) ( ) q 1 g g (mod q) q g g 2 = ( ) ( ) q 1 (mod q) q ( ) ( ) q 1 (mod q) q

±1 q q 3 ( q ) ( ) q = 1 = ( 1) 1 2 ( ) ( ) q = ( 1) 1 2 q 1 2 q 6.2 3 6.1. π N(π) 3 1, ω, ω 2 mod π 6.2. π N(π) 3 π /α Z[ω] α N(π) 1 3 ω m (mod π) m = 0, 1, 2. α N(π) 1 1 = (α N(π) 1 3 1)(α N(π) 1 3 ω)(α N(π) 1 3 ω 2 ) 0 (mod π)( 4.2(k)) 4.2(h) ( 6.3 (3 ). π N(π) 3 α ) α Z[ω] π 3 ( α π ) 3 = { 0 (π α) ω m ( α ) 6.4 ( ). () π 3 (b) ( ) αβ ( α ) = π 3 π 3 (π /α) ω m 6.2 ( α π ) ( ) β π 3 3 = 1 x 3 α (mod π) (c) ( α 1 α ) 3 (N(π) 1) π 3 (mod π)

( α ) (d) α β (mod π) = π 3 ( ) β π 3 (c) Euler 3.2(d) 10( α ) χ π (α) π 3 6.5. χ π (α) = χ π (α) 2 = χ π (α 2 ) χ π (α) = χ π (α). α 1 3 (N(π) 1) χ π (α) (mod π) α 1 3 (N(π) 1) χ π (α) (mod π) N(π) = N(π) χ π (α) = χ π (α) 6.6. q Z[ω] χ q (α) = χ q (α 2 ) (n, q) = 1 n χ q (n) = 1. q = q, n = n 6.5 6.7 (rimry). π Z[ω] π 2 (mod 3) π rimry π π rimry 11 π q π = q rimry ππ = π = + bω( 2, b 0 (mod 3) rimry π 12 10 11 12

1: mod3 0 1 2 ω 1 + ω 2 + ω 2ω 1 + 2ω 2 + 2ω. mod3 13 1 ω π 2ω (mod 3) π 2ω, π ω, ωπ 1+ω, ωπ 2+2ω, ω 2 π 2, ω 2 π 1 (mod 3) π rimry π π rimry 6.3 6.8 ( ). π 1, π 2 rimry N(π 1 ) N(π 2 ) N(π 1 ), N(π 2 ) 3 χ π1 (π 2 ) = χ π2 (π 1 ) 6.9 ( ). π N(π) 3 π = q q = 3m 1 ππ = π rimry = 3m 1 χ π (1 ω) = ω 2m 6.4 6.4.1 π 1 = q 1, π 2 = q 2 6.6 χ π1 (π 2 ) = χ π2 (π 1 ) = 1 13

6.4.2 π 1, π 2 ππ = ππ = Z[ω]/πZ[ω] Z/Z = F χ π F 3 g(χ π ) 3 5.13 J(χ π, χ π ) χ π J(χ π, χ π ) Z[ω] 6.10. J(χ π, χ π ) π π. J(χ π, χ π ) = x 5.11 x 2 = xx = = ππ Z[ω] x π x π 6.11. J(χ π, χ π ) 2 (mod 3). g(χ π ) 3 = ( t χ π (t)ζ t ) 3 t χ π (t) 3 ζ 3t (mod 3) 2.6 = 1 (χ π (0) 3 = 0, t 0 χ π (t) 3 = 1 ) g(χ π ) 3 = J(χ π, χ π ) 1 (mod 3) 6.12. n 0 (mod N(π) 1). t 0 (mod ) n (t) n (mod ) = t n n F n 0 (mod π) t 0 (mod )

6.13. J(χ π, χ π ) = π. 6.10 6.11 π rimry ππ = J(χ π, χ π ) = π π J(χ π, χ π ) = χ π ()χ π (b) +b=1 = χ π ((1 )) ( 2 ) 1 3 ( 1) (mod π) = 2 3 ( 1) n= 1 3 ( 1) k n n (k n ) 0 (mod π) ( 6.12) J(χ π, χ π ) 0 (mod π) π 0 (mod π) J(χ π, χ π ) = π g(χ π ) 3 = π 6.4.3 π 1 π 1 =, π 2 = q. g(χ π ) 3 = π g q2 1 = (π) 1 3 (q2 1) χ q (π) (mod q) g q2 χ q (π)g (mod q) g q2 = ( t χ π (t)ζ t ) q2 ( t t χ π (t) q ζ tq ) q (mod q) ( 2.6) χ π (t)q 2 t tq2 (mod q) q 2 1 (mod 3) = t χ π (t)t tq2 = g q 2 g q 2 = χ π ((q 2 ) 1 )g χ π ((q 2 ) 1 )g χ q (π)g (mod q)

g(χ π ) g(χ π )g(χ π ) = χ π ( 1) χ π ((q 2 ) 1 ) χ q (π) (mod q) 6.6 χ q () = 1, χ π ((q 2 ) 1 ) = χ π (q), 1, ω, ω 2 mod q χ π (q) = χ q (π) 6.4.4 π 1 π 1 = 1, π 2 π 2 = 2. 6.4.3 χ π1 ( 2 2) = χ π2 (π 1 1 ) χ π2 ( 2 1) = χ π1 (π 2 2 ) 14 χ π1 (π 2 )χ π2 (π 1 ) = 1 χ π (α) = χ π (α) χ π1 (π 2 ) = χ π2 (π 1 ) 6.5 π = q ππ ( = α ) τ τ 3 6.6 Z[ω] 14 /π = π

L A TEX s.genki0605@gmil.com [1] G. H. Hrdy, E. M. Wright I, 2001 [2] J. H. Silvermn,2007 [3] T. M. Aostol Introduction to Anlytic Number Theory (Sringer,1976) [4],1992

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