2014 (2014/04/01)

2014 (2014/04/01)

1 5 1.1...................................... 5 1.2...................................... 7 1.3...................................... 8 1.4............................... 10 1.5 Zorn........................... 11 1.6.................................... 12 1.7................................ 13 2 17 2.1................................. 17 2.2...................................... 19 2.3..................................... 20 2.4..................................... 23 2.5..................................... 25 3 27 3.1.................................... 27 3.2...................... 29 3.3..................................... 31 3.4............................... 32 3.5.................................... 34 3.6.................................... 38 3

Chapter 1 n V V ( ) n R R R R n R,, [1] 1.1 A (set) a A (element) a A A a a A a A B A 5

6 CHAPTER 1. (subset) B A B = A B A B A B A (proper subset) B A (empty set) B A A B = {a A a B} A (finite set) A A A (infinite set) A = A < A. n {a 1, a 2,, a n } {a 1, a 2, } A B, A B (intersection) (union) A i (i = 1, 2,, n) n n A i, i=1 A i (i = 1, 2, ) A i, A i Λ i=1 i=1 A λ (λ Λ) λ Λ A λ, Λ a [ a, a] [ a, a] i=1 λ Λ A i A λ a {b R b>0} a>0[ a, a] 1.1.1. a>0 [ a, a] a>0[ a, a] A λ λ λ A λ A λ = λ Λ (disjoint union) A λ A λ < λ Λ λ Λ A λ < A λ = A λ λ Λ λ Λ A B A B (a, b) A B A B (direct product, cartesian product) A B = {(a, b) a A, b B} λ Λ

1.2. 7 A λ (λ Λ) λ Λ A λ (Λ (Zermelo s axiom of choice) ) 1.2 N : Z : ( ) Q : () R : ( ) C : () N 0 a, b Z l Z b = al b a a b a b a b (divisor) b a (multiple) 0 1 0 a λ (λ Λ) λ Λ c a λ c Z a λ (λ Λ) (common divisor) (greatest common divisor) a 1, a 2, (a 1, a 2, ) gcd(a 1, a 2, ) gcd(a, b) = 1 a b p N, p > 1 p (prime number) p 1 p p ab p a p b n N a, b Z n a b a b n (congruent modulo n) a b (mod n) 1.2.1. (1) a Z a a (mod n) (2) a b (mod n) b a (mod n) (3) a b (mod n) b c (mod n) a c (mod n) ( n Z )

8 CHAPTER 1. 1.3 A B A B (map) A B f f : A B f a A B f a f(a) f : A B (a f(a)) f : A B A f (domain) B f (range) f : A B g : C D A = C B = D a A f(a) = g(a) f = g f : A B f(a) = Imf = {f(a) a A} f (image) C A f(c) = {f(a) a C} f C f : A B C B f 1 (C) = {a A f(a) C} f C (inverse image) C = {b} f 1 ({b}) f 1 (b) f 1 (b) = {a A f(a) = b} b f(a) f 1 (b) = f 1 (b) f 1 B A f : A B (injection) a a f(a) f(a ) f : A B (surjection) f(a) = B f : A B (bijection) f 1.3.1. f : A B (1) f (a a f(a) f(a ) ) (2) f(a) = f(a ) a = a (3) b f(a) f 1 (b) = 1 (4) b B f 1 (b) 1 1.3.2. f : A B (1) f (f(a) = B )

1.3. 9 (2) b B f(a) = b a A (3) b B f 1 (b) 1 B A ι : B A (b b) B A (inclusion) B = A ι : A A (a a) A (identity map) id A f : A B g : B C A C (a g(f(a))) f g (composite map) g f gf f : A B b B f(a) = b a A f 1 (b) = {a} f 1 (b) a A B A (b f 1 (b)) f (inverse map) f 1 ( ) f 1 f f 1 = id B, f 1 f = id A, (f 1 ) 1 = f f : A B C A C g : C B (c f(c)) f C (restriction) f C ι : C A f : A B f ι 1.3.3. f : Z Z (1) f (2) f (3) f f(0) = 1 f(1) = 1 1.3.4. A < f : A A (1) f (2) f (3) f 1.3.5. f : A B g : B C (1) g f g (2) g f f 1.3.6. f : A B g : B A g f f g f

10 CHAPTER 1. 1.3.7. f : A B g : B C g f (g f) 1 = f 1 g 1 1.3.8. f : A B C A f C f f f C f : A B a A f(a) B A < a A f(a) A = {1, 2, 3}, B = {a, b} 1 2 3 a a b f(1) = a, f(2) = a, f(3) = b f : A B 1.3.9. A = m <, B = n < A B 1.4 A R A A R A ( ) (binary relation) (a, b) R arb A (E1) [ ] a A a a (E2) [ ] a b b a (E3) [ ] a b b c a c (equivalence row) (equivalence relation) a b a b ( ) A a A C a = {b A b a} a (equivalence class) 1.4.1. A (1) a C a (2) b C a a C b (3) C a C b C a C b =

1.5. ZORN 11 {C λ λ Λ} A = λ Λ C λ, (λ µ C λ C µ = ) A C λ a λ a λ C λ {a λ λ Λ} A = λ Λ C λ A A/ 1.4.2. 1.2.1 a b (mod n) Z 1.4.3. n M n (R) A, B M n (R) P B = P 1 AP A B M n (R) 1.4.4. A, B M n (R) P B = AP A B M n (R) 1.4.5. f : A B A f(a) = f(a ) a a 1.5 Zorn A (O1) [ ] a A a a (O2) [] a b b a a = b (O3) [ ] a b b c a c (order) (A, ) (ordered set) A a b b a a b a b a b a < b B A B A 1.5.1. R Q, Z R R (A, ) a, b a b b a (totally order) (A, ) (totally ordered set) ( (partially order) ) 1.5.2. A P (A) P (A) A (power set) 2 A P (A) A 2 P (A)

12 CHAPTER 1. (A, ) a b b A a b, b A a = b a A (maximal element) b a b A a A (minimal element) b A b a a A (largest element) b A a b a A (smallest element) ( ) ( ) ( ) 1.5.3. (0, 1) (0, 1) ( ) ( ) 1.5.4. A P (A) S = {X P (A) X A} a A A {a} S S = {X P (A) X } a A {a} S B A a A B b B b a B B A A 1.5.5 (Zorn ). A A Zorn (A, ) (well ordered set) A 1.6 A f : A A A A ( ) f (a, b) f(a, b) ab a + b ab ab a + b a + b a, b, c A (ab)c = a(bc) ab = ba a b A a, b, c A a(b + c) = ab + ac, (a + b)c = ac + bc

1.7. 13 1.6.1. (1) Z Q, R, C (2) Z (3) n 2 R n M n (R) M n (R) A f : A A A (a, b) A A f(a, b) A A < A = {a, b, c} a b c a a b c b c a b c b c a f(b, a) = c 1.6.2. 1.7 A ( ) (ab)c = a(bc) A (semigroup) A n a 1, a 2,, a n (( ((a 1 a 2 )a 3 ) )a n 1 )a n a 1 a 2 a n 3 3 1.7.1 ( ). A n. n n 3 n 4 n 1 XY X r Y n r r = n 1 X = a 1 a 2 a n 1 XY = a 1 a 2 a n r n 2 X = a 1 a 2 a r, Y = a r+1 a r+2 a n XY = (a 1 a 2 a r )(a r+1 a r+2 a n ) = (a 1 a 2 a r )((a r+1 a r+2 a n 1 )a n ) = ((a 1 a 2 a r )(a r+1 a r+2 a n 1 ))a n = (a 1 a 2 a n 1 )a n = a 1 a 2 a n 1 a n

14 CHAPTER 1. A ab = ba A n A e a A ae = ea = a e A (identity element) (monoid) 1.7.2. (1) N ( ) 1 (2) N {1} (3) Z ( ) 0 (4) N 1.7.3.. e, e e e = ee e e = ee e = e A 1 1 A 0 0 A ( 1 1 A 1 ) A a n a 0 = 1 A, a n = a n 1 a a n a n (a to the n-th power) 1.7.4. A a A m, n N (1) a m a n = a m+n (2) (a m ) n = a mn (3) ab = ba (ab) m = a m b m 1.7.5. X X X X X σ, τ X X στ (στ)(x) = σ(τ(x)) (στ = σ τ ) X X id X A u A uu = u u = 1 u A u A (unit) u u (inverse element)

1.7. 15 1.7.6. A u. u, u u u = u 1 = u (uu ) = (u u)u = 1u = u u u 1 u 1 (u 1 ) 1 = u 1.7.7. A 1 A (1 A ) 1 = 1 A 1.7.8. u 1, u 2,, u n u 1 u 2 u n (u 1 u 2 u n ) 1 = u n 1 u 2 1 u 1 1 u A 0 n N u 0 = 1 A, u n = (u 1 ) n m, n Z 1.7.9. X X ( 1.7.5 ) σ X X σ

Chapter 2 2.1 (group) G (G1) [] a, b, c G a(bc) = (ab)c (G2) [ ] e G a G ea = ae = a ( e 1 G ) (G3) [ ] a G b G ab = ba = e ( b a 1 ) G G (G4) [] a, b G ab = ba G (abelian group) (commutative group) 2.1.1. G (1) [] ax = ay x = y xa = ya x = y (2) f : G G (x x 1 ) (3) a G g a : G G (x xa) h a : G G (x ax) k a : G G (x a 1 xa) 17

18 CHAPTER 2.. (1) ax = ay a 1 x = y (2) (x 1 ) 1 = x f 2 = id G f (3) g a g a 1 = g a 1 g a = id G g a 2.1.2. G x G x 2 = 1 G. x G x 2 = 1 x 1 = x a, b G (ab) 1 = ab (ab) 1 = b 1 a 1 = ba ab = ba 2.1.3. Q = Q {0} Q 1 a Q 1/a R = R {0}, C = C {0} 2.1.4. (1) Q 0 (2) Z {0} 2 2.1.5. M U M U M. a, b U ab U U M U U 1 U U a U a 1 U U U U(M) M (unit group) 2.1.6. (1) Q U(Q) = Q = Q {0} (2) Z U(Z) = { 1, 1} 2.1.7 ( ). X X ( 1.7.5) U(X X ) X (symmetric group) S(X) S(X) X X X (permutation) ( ) x S(X) σ = σ(x) X = n X = {1, 2,, n} S(X) S n n S n n 2.1.8. 3 S 3 ( ) ( ) ( ) 1 2 3 1 2 3 1 2 3,,, 1 2 3 1 3 2 2 1 3 S 3 = ( ) ( ) ( ) 1 2 3 1 2 3 1 2 3,, 2 3 1 3 1 2 3 2 1

2.2. 19 ( 1 2 3 3 1 2 ) ( 1 2 3 1 3 2 ) = ( 1 2 3 3 2 1 1 1 1 3 ( ) 1 ( ) ( ) 1 2 3 2 3 1 1 2 3 = = 2 3 1 1 2 3 3 1 2 2.1.9. S 3 2.1.10. n 3 S n στ τσ σ, τ S n G G < G (finite group) G = G (infinite group) G < G G (order) 2.1.11. n S n n! 2.1.12 ( ). R n M(n, R) M(n, R) R n (general linear group) GL(n, R) M(n, R) GL(n, R) GL(n, R) GL(n, Q), GL(n, C) ( M n (R), GL n (R) ) 2.1.13. n 2 GL(n, R) 2.1.14. A x, y, z A xz = yz x = y A A ) 2.2 G G (additive group) 0 0 G a a (A1) [] a, b, c G a + (b + c) = (a + b) + c

20 CHAPTER 2. (A2) [ ] 0 G a G 0 + a = a + 0 = a (A3) [ ] a G b G a + b = b + a = 0 ( b a ) (A4) [] a, b G a + b = b + a G a + ( b) a b 2.2.1. Z, Q, R, C ( ) N n N G a n na a n na 0a = 0 m Z ma a, b G, m, n Z (1) ( m)a = m( a) = (ma) ( 1)a = a (2) (m + n)a = ma + na (3) m(na) = (mn)a (4) m(a + b) = ma + mb (2), (3), (4) 2.3 G H (B1) a, b H ab H (B2) a H a 1 H H G (subgroup) 2.3.1. G H (1) H G (2) H G (3) a, b H ab 1 H. (1) = (2) H G (B1) a, b H ab H G H G H H a (B2) a 1 H 1 G = aa 1 H 1 G H (B2)

2.3. 21 (2) = (3) (3) = (1) H a H 1 = aa 1 H a H 1 H a 1 = 1a 1 H (B2) a, b H (B2) b 1 H ab = a(b 1 ) 1 H (B1) 2.3.2. H, K G H K G. a, b H K ab 1 H K a H, b H H ab 1 H K ab 1 K ab 1 H K G A, B AB = {ab a A, b B} A 1 = {a 1 a A} B = {b} A{b} Ab ba Ab = {ab a A}, ba = {ba a A} 2.3.3. G A, B, C (1) A(BC) = (AB)C (2) (A 1 ) 1 = A (3) (AB) 1 = B 1 A 1 2.3.4. G H (1) H G (2) HH H H 1 H (3) HH 1 H 2.3.5. H G HH = HH 1 = H 1 = H. HH 1 = 1 ( ) 2.3.6. G H H < HH H H G

22 CHAPTER 2.. h H h 1 H HH H h 2 H n N h n H H h n m, n N, m < n h m = h n h n m = 1 n m = 1 1 = h H h 1 = 1 H n m > 0 n m 1 0 h 1 = h n m 1 H 2.3.7. H, K G (1) HK G HK = KH (2) L H G (HK) L = H(K L). (1) HK G (HK) 1 = HK H 1 = H, K 1 = K (HK) 1 = K 1 H 1 = KH HK = KH HK = KH (HK)(HK) 1 = HKK 1 H 1 = HKKH = HHKK = HK HK G (2) x HK L x HK h H k K x = hk x L h H L k = h 1 x L k K L x = hk H(K L) (HK) L H(K L) y H(K L) h H k K L y = hk y = hk HK h H L y = hk L y HK L H(K L) HK L (HK) L = H(K L) G G {1} G {1} G (trivial subgroup) 1 G (proper subgroup) S G a 1 n 1 a 2 n2 a r n r (a i S, n i Z, r N) G S S (subgroup generated by S) S {s 1,, s l } S s 1, s l S = {a} a = {a n n Z} = {, a 2, a 1, 1, a, a 2, } a (cyclic group) a (generater) a a (order) o(a) 2.3.8. S 2.3.9. a

2.4. 23 (1) a m = 1 m N a (2) a a m = 1 m N n n = o(a) (i) a m = 1 n m (ii) a = {1, a, a 2,, a n 1 } (3) a, a 2, a 1, 1, a, a 2, a. (1) a m = 1 m N l Z l = nq + r, 0 r < m q, r Z a l = (a m ) q a r = a r a {1, a, a 2,, a m 1 } a 0 < s < t a s = a t a t s = 1, t s N (2) (i) a m = 1 m = nq + r, 0 r < n q, r Z 1 = a m = (a n ) q a r = a r n r = 0 n m n m a m = (a n ) m/n = 1 (i) a = {1, a, a 2,, a n 1 } 0 i < j < n a i = a j a j i = 1, 0 < j i < n n o(a) = a = n (2), a 2, a 1, 1, a, a 2, i < j (i, j Z) a i = a j a j i = 1, 0 < j i a 2.4 H G G ah = bh a b G a G ah = ah a a a b ah = bh bh = ah b a a b b c ah = bh = ch a c 2.4.1. H G H a, b G (1) a b ( ah = bh) (2) b ah

24 CHAPTER 2. (3) a bh (4) a 1 b H. (1) = (2) b = b1 bh = ah (2) = (3) b ah h H b = ah h 1 H a = bh 1 bh (3) = (4) a bh h H a = bh a 1 b = h 1 H (4) = (1) h H a 1 b = h a = bh 1, b = ah h 1 H ah 1 = bh 1 h 1 bh ah bh h 2 H bh 2 = ahh 2 ah bh ah ah = bh Ha = Hb ah H (left coset) G/H Ha H (right coset) H\G G = i I a i H 2.4.2. G = i I a ih G = i I Ha i 1 2.4.3. 3 S 3 S 3 ( ) ( ) ( ) 1 2 3 1 2 3 1 2 3 g 1 =, g 1 2 3 2 =, g 1 3 2 3 =, ( ) ( ) ( 2 1 3 ) 1 2 3 1 2 3 1 2 3 g 4 =, g 2 3 1 5 =, g 3 1 2 6 = 3 2 1 H = g 2 = {g 1, g 2 } g 1 H = g 2 H = {g 1, g 2 } g 3 H = g 4 H = {g 3, g 4 } g 5 H = g 6 H = {g 5, g 6 } Hg 1 = Hg 2 = {g 1, g 2 } Hg 3 = Hg 5 = {g 3, g 5 } Hg 4 = Hg 6 = {g 4, g 6 }

2.5. 25 K = g 4 = {g 1, g 4, g 5 } g 1 K = g 4 K = g 5 K = {g 1, g 4, g 5 } g 2 K = g 3 K = g 6 K = {g 2, g 3, g 6 } Kg 1 = Kg 4 = Kg 5 = {g 1, g 4, g 5 } Kg 2 = Kg 3 = Kg 6 = {g 2, g 3, g 6 } H ah = Ha a G H G (normal subgroup) G 2.4.4 (Lagrange). G H a G ah = H G : H G = G : H H. a G f : H ah f(h) = ah ah = H G = i I a ih G = i I a i H = G : H H G : H G H (index) 2.4.5. G x G x G G x x G = 1 2.5 G N a G an = Na G/N (an)(bn) = (ab)n

26 CHAPTER 2. an = a N a G a ( ) an = a N bn = b N (ab)n = (a b )N an = a N bn = b N n 1, n 2 N a = an 1, b = bn 2 bn = Nb n 3 N n 1 b = bn 3 a b = an 1 bn 2 = abn 3 n 2 (ab)n (a b )N = (ab)n (1N)(aN) = (an)(1n) = an, (an)(a 1 N) = (a 1 N)(aN) = 1N G/N 1N an a 1 N G N (factor group) G/N 2.5.1. Z n N n n n nz a Z nz a + nz = {a + nl l Z} {0, 1,, n 1} Z/nZ = {0 + nz, 1 + nz,, (n 1) + nz} (3 + 5Z) + (4 + 5Z) = 7 + 5Z = 2 + 5Z

Chapter 3 3.1 R R (ring) (R1) R (R2) R (R3) [] a, b, c R a(b + c) = ab + ac, (a + b)c = ac + bc (R4) [ ] 1 R ( 0) R (R1), (R2), (R3) (R5) [] a, b R ab = ba R (commutative ring) R 0 0 R R 1 1 R 3.1.1. R x 0x = x0 = 0 ( 0 R 0 R 0 Z Z 0 R x 0 Z x ) R R U(R) U(R) R (unit group) R (unit) ( ) R 0 R (skew field, division ring) (field) (commutative field) 27

28 CHAPTER 3. 3.1.2 ( ). a a + a = a, aa = a 3.1.3 ( ). Z (rational integer ring) 3.1.4 ( ). Q, R, C (rational number field) (real number field) (complex number field) 3.1.5 (). R () R n R n (full matrix ring) M(n, R) M n (R) R M(n, R) R 0 a R R (left zero divisor) 0 b R ab = 0 0 a R R (right zero divisor) 0 b R ba = 0 0 ( ) 3.1.6. R ( ) R R ( ). a 0 b R ab = 0 b = 1b = a 1 ab = a 1 0 = 0 b 0 R a a (zero divisor) R (integral domain) R 3.1.7. Z 3.1.8. A A 3.1.9. Z M(n, Z) ax = ay x = y a 0 3.1.10. R 0 a R, x, y R ax = ay x = y. ax = ay a(x y) = 0 a 0 x y = 0 x = y 3.1.11. C M(2, C) ax = ay x y

3.2. 29 3.2 n N, n 2 a, b Z l Z a b = nl a b (mod n) ( 1.2.1) a a + nz = {b Z a b (mod n)} = {a + nl l Z} Z/nZ = {0 + nz, 1 + nz,, (n 1) + nz} 2.5.1 Z/nZ Z nz (a + nz) + (b + nz) = (a + b) + nz (a + nz)(b + nz) = ab + nz Z/nZ a + nz = a + nz, b + nz = b + nz l, l Z a = a + nl, b = b + nl a b = (a + nl)(b + nl ) = ab + n(al + bl + nll ) ab + nz a b + nz = ab + nz Z/nZ 1 + nz Z/nZ n a + nz Z/nZ ā Z/nZ 3.2.1. Z/9Z 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 2 0 2 4 6 8 1 3 5 7 3 0 3 6 0 3 6 0 3 6 4 0 4 8 3 7 2 6 1 5 5 0 5 1 6 2 7 3 8 4 6 0 6 3 0 6 3 0 6 3 7 0 7 5 3 1 8 6 4 2 8 0 8 7 6 5 4 3 2 1 1 0 0 U(Z/9Z) = { 1, 2, 4, 5, 7, 8} 3, 6

30 CHAPTER 3. 3.2.2. a, b N gcd(a, b) = d x, y Z ax + by = d. a > b b b = 1 gcd(a, b) = 1 x = 0, y = 1 b > 1 a = bq + r, 0 r < b q, r Z gcd(a, b) = gcd(b, r) d b d a d a bq = r d r d bq + r = a d a, b b, r gcd(a, b) = gcd(b, r) r = 0 gcd(a, b) = gcd(b, 0) = b x = 0, y = 1 d = gcd(a, b) 0 < r b > r b, r x, y Z bx + ry = d d = bx + ry = bx + (a bq)y = ay + b(x qy ) x = y, y = x qy Z/nZ 3.2.3. a + nz Z/nZ gcd(a, n) = 1 U(Z/nZ) = {a + nz gcd(a, n) = 1}. gcd(a, n) = 1 3.2.2 x, y Z ax+ny = 1 n ā x = 1 ā ā b Z ā b = 1 l Z ab 1 = nl 1 = ab nl gcd(a, n) 1 gcd(a, n) gcd(a, n) = 1 Z/nZ 3.2.4. 0 ā Z/nZ (1) ā (2) ā

3.3. 31 (3) gcd(a, n) > 1. (2) (3) 3.2.3 (1) = (2) (3) = (1) gcd(a, n) = d > 1 n = dl 1 < l < n a = da l 0 ā l = ā n = 0 ā 3.2.5. Z/nZ (1) Z/nZ (2) Z/nZ (3) n. (1) (2) (3) = (1) n 1 a < n gcd(a, n) = 1 ā Z/nZ (1) = (3) Z/nZ 1 a < n gcd(a, n) = 1 n 3.2.6. 128x + 405y = 1 (x, y) 3.2.7. M 2 (Z/2Z) 3.3 R R S R (subring) a, b S a b S, ab S S R S 3.3.1. Z C, R, Q 3.3.2. R = M(n, R) S = {(a ij ) R i > j a ij = 0} = a 11 a 12 a 1n a 22 a 2n.... 0 a nn a ij R S R A = (a ij ), B = (b ij ) S A B S AB S AB = (c ij ) n c ij = a ik b kj k=1 i > j i > k a ik = 0 k > j b kj = 0 i k j a ik b kj 0 i > j k a ik b kj = 0 c ij = 0 AB S

32 CHAPTER 3. S R R S 3.3.3. Z/6Z S = { 0, 2, 4} S Z/6Z 1 S 4 3.3.4. R = M(2, R) { ( a b S = b a ) } a, b R R 3.4 R R I R (ideal) (I1) i, j I i j I (I2) a R, i I ai I (I3) a R, i I ia I (I1), (I2) I (left ideal) (I1), (I3) I (right ideal) ( ) (two-sided ideal) I R a, b R a b (mod I) a b I R a R a + I = {a + i i I} R/I a + I, b + I R/I (a + I) + (b + I) = (a + b) + I, (a + I)(b + I) = ab + I

3.4. 33 a + I = a + I, b + I = b + I a = a + i, b = b + j i, j I a b ab = (a + i)(b + j) ab = ib + aj + ij (I3) ib I (I2) aj I (I2) ij I (I1) ab = ib + aj + ij I a b + I = ab + I 0 + I a + I a + I R/I R/I R I (factor ring) R R/I R 1 I R R/I 1 + I 3.4.1. n N nz = {nl l Z} Z ( Z/nZ ) 3.4.2. R = { ( a b 0 c ) } { ( 0 b a, b, c R, I = 0 0 ) } b R I R R R {0 R } R R (trivial ideal) 3.4.3. R I I = R 1 R I 3.4.4. R a R ar = {ar r R} R ( ar a (principal ideal) ) 3.4.5. R a R {r 1 ar 2 r 1, r 2 R} R a 3.4.6. Z ( (principal ideal domain) )

34 CHAPTER 3. 3.5 R R x f(x) = a 0 + a 1 x + + a n x n (a i R) x R (polynomial) f(x) f x (indeterminate) x R R[x] R[x] f(x) = a 0 + a 1 x + + a n x n, g(x) = b 0 + b 1 x + + b m x m l f(x) + g(x) = (a i + b i )x i l = max(n, m) 0 i=0 f(x)g(x) = n+m k=0 c k x k, c k = i+j=k a i b j R[x] x R (polynomial ring) f(x) = a 0 + a 1 x + + a n x n a n 0 n f(x) (degree) deg f(x) deg f f(x) = 0 deg 0 = d d = d, + d = R 3.5.1. R f(x), g(x) R[x] deg(f + g) max(deg f, deg g) deg(fg) = deg f + deg g. f(x) 0, g(x) 0 f(x) = m i=0 a ix i (a m 0), g(x) = n j=0 b jx j (b n 0) f(x)g(x) = m+n m k=0 i=0 a ib k i x k x m+n a m b n a m 0 b n 0 R a m b n 0 f(x) = 0 g(x) = 0 n {, 0} N +n = 3.5.2. R R[x]

3.5. 35. R[x] R[x] f(x), g(x) R[x], f(x) 0, g(x) 0 deg(f), deg(g) 3.5.1 deg(fg) f(x)g(x) 0 deg f(x) = 0 f(x) f(x) = 0 R R R[x] 3.5.3. R deg(fg) < deg f + deg g f(x), g(x) R[x] 3.5.4. R f(x), g(x) R[x] g(x) R q(x), r(x) R[x] f(x) = g(x)q(x) + r(x), deg r < deg g. f(x) = a 0 + a 1 x + + a n x n, g(x) = b 0 + b 1 x + + b m x m q(x), r(x) n = deg f g(x) 0 deg g 0 n = n < m q(x) = 0, r(x) = g(x) n m h(x) = f(x) a n b m 1 x n m g(x) deg h < n h(x) = g(x)q 1 (x) + r(x), deg r < deg g q 1 (x), r(x) R[x] f(x) = h(x) + a n b m 1 x n m g(x) = g(x)(q 1 (x) + a n b m 1 x n m ) + r(x) f(x) = g(x)q(x) + r(x) = g(x)q (x) + r (x), deg r, deg r < deg g g(x)(q(x) q (x)) = r (x) r(x) q(x) q (x) deg g deg g q(x) = q (x) r(x) = r (x) R g(x) 1

36 CHAPTER 3. 1 (monic) 3.5.4 q(x), r(x) f(x) g(x) r(x) = 0 f(x) g(x) g(x) f(x) f(x) = a 0 + a 1 x + + a n x n R[x] α R f(α) = a 0 + a 1 α + + a n α n R f(x) α f(α) = 0 α f(x) (root) 3.5.5. R f(x) R[x], α R (1) [] q(x) R[x] f(x) = (x α)q(x) + f(α) (2) [] f(α) = 0 x α f(x). f(x) g(x) = x α 3.5.4 q(x), r(x) R[x] f(x) = (x α)q(x) + r(x), deg r < deg(x α) = 1 r(x) = r R α f(α) = r 3.5.6. K K[x] ( 3.4.6 ) 3.5.7. R 0 f(x) R[x], deg f = n f(x) n. n n = 0 f(x) = r 0 0 n 1 f(x) f(x) α f(x) = (x α)g(x) g(x) R[x] deg g = n 1 g(x) n 1 β f(x) 0 = f(β) = (β α)g(β) R β α = 0 g(β) = 0 f(x) α g(x) f(x) n f(x) R[x] f : R R (α f(α))

3.5. 37 3.5.8. f(x), g(x) R[x] R f(x) = g(x) f = g. f(x) = g(x) f = g f(x) g(x) h(x) = f(x) g(x) h(x) 0 h(x) deg h R h(α) 0 α R 0 h(α) = f (α) g (α) f g 3.5.9. p R = Z/pZ R f(x) = x p x α R f(α) = 0 f = 0 f(0) = 0 α 0 α U(Z/pZ) U(Z/pZ) p 1 2.4.5 α p 1 = 1 f(α) = α p α = 0 R[x 1, x 2,, x n ] R[x 1, x 2,, x n ] = R[x 1, x 2,, x n 1 ][x n ] (R[x 1,, x n 1 ] x n ) f(x 1, x 2,, x n ) = a i1 i 2 i n x 1 i 1 x 2 i2 x n i n, (a i1 i 2 i n R) x 1, x 2,, x n R (polynomial) a i1 i 2 i n 0 a i1 i 2 i n x 1 i 1 x 2 i2 x n i n f (term) i 1 + i 2 + + i n (degree) 3.5.10. R R[x 1, x 2,, x n ] R. R R[x 1 ] R[x 1, x 2 ] = R[x 1 ][x 2 ] R[x 1, x 2,, x n ] U(R) = U(R[x]) R[x 1, x 2,, x n ] R f(x) R[x] g(x) R[x] f(x)g(x) = 1 deg f + deg g = 0 deg f = deg g = 0 f(x), g(x) R f(x) R U(R[x]) U(R) U(R) U(R[x]) U(R) = U(R[x]) f(x 1, x 2,, x n ) R[x 1, x 2,, x n ] f : R R R R 3.5.11. R f(x 1, x 2,, x n ), g(x 1, x 2,, x n ) R[x 1, x 2,, x n ] f g f g. f = g f = g f g f 0 f 0 n n = 1 f R[x 1,, x n 1 ] x n m i f(x 1,, x n ) = g i (x 1,, x n 1 )x n i=0

38 CHAPTER 3. f 0 i g i 0 g i (α 1,, α n 1 ) 0 (α 1,, α n 1 ) R R 0 f(α 1,, α n 1, x n ) R[x n ] α n R f(α 1,, α n 1, α n ) 0 f 0 3.6 K 1 K 1, 1 + 1, 1 + 1 + 1, 1, 2, 3, 0, 1, 2 = ( 1) + ( 1), F = {, 2, 1, 0, 1, 2, } F K K F F F 1 Z/nZ (n N) Z ( 2.3.9) F = Z/nZ F n ( 3.2.5) K (characteristic) F = Z K 0 p ( 0) p = 0 0 3.6.1. Q, R, C 0 Z/pZ (p ) p 3.6.2. K p ( 0) a, b K (a + b) p = a p + b p. (a + b) p = p i=0 ( ) p a i b p i 0 < i < p i ( ) p = i p! i!(p i)! p p p K 0 3.6.3. F = Z/5Z n N f n : F F f(a) = a n n = 2, 3, 4, 5 f n ( ) 3.6.4. p F = Z/pZ a F a p = a

3.6. 39 3.6.5 ( Q ). Z Q Z = Z {0} ( ) Z Z at = bs (a, s) (b, t) (a, s) a/s (Z Z )/ R R a/s + b/t = (at + bs)/st (a/s)(b/t) = (ab)/(st) R 1/1 0/1 a/s (a 0) s/a R Q 3.6.6. 3.6.5 (1) (2) (3) 3.6.5 Z D D (quotient field) 3.6.7. R R R[x] R[x] { } f(x) g(x) g(x) 0 R R(x) R K K(x) = R(x) m Z m = a 2 a Z m Z (square free) m 0, 1 m 1 m m 3.6.8. 3, 15, 6, 105 0, 1, 4, 9, 12 m Q[ m] = {a + b m a, b Q} Q( { a + b } m m) = c + d m a, b, c, d Q, c2 + d 2 0

40 CHAPTER 3. 3.6.9. Q[ m]. R = Q[ m] C 1 R α, β R α β, αβ R R 0 a+b m (a, b Q) (a+b m)(a b m) = a b b 2 m m 0 a + b m C R 1 a + b m = a b m (a + b m)(a b m) = a a 2 b 2 m R = Q[ m] 3.6.10. Q[ m] = Q( m) b m Q[ m] a 2 b 2 m Q[ m] (quadratic field) Q[x] (x 2 m)q[x] Q[x]/(x 2 m)q[x] f(x) Q[x] ( ) Q[x]/f(x)Q[x] (algebraic number field) R f(x) R[x] (irreducible) g(x), h(x) R[x] f(x) = g(x)h(x) g(x) h(x) R[x] ( R ) (reducible) K f(x) K[x] (K ) ( ) f(x), g(x) K[x] f(x) g(x) 3.6.11. K f(x) g(x) h(x), l(x) K[x] f(x)h(x) + g(x)l(x) = 1 3.6.12. K f(x) K[x] K[x]/f(x)K[x] (K = Q, deg f(x) = n n ). 0 g(x) K[x]/f(x)K[x] g(x) g(x) 0 g(x) f(x) f(x) f(x) g(x) 3.6.11 f(x)h(x) + g(x)l(x) = 1 h(x), l(x) K[x] K[x]/f(x)K[x] g(x) l(x) = 1 g(x) K[x]/f(x)K[x]

3.6. 41 Q n n f(x) = a 0 + a 1 x + + a n x n Z[x] 1 f(x) Z[x] a g(x) f(x) = ag(x) 3.6.13. f(x) = m i=0 a ix i, g(x) = n j=0 b jx j Z[x] f(x)g(x). p f(x) g(x) p a 0, p a 1,, p a i 1, p a i, p b 0, p b 1,, p b j 1, p b j i, j f(x)g(x) x i+j a 0 b i+j + a 1 b i+j 1 + + a i 1 b j+1 + a i b j + a i+1 b j 1 + a i+j b 0 p f(x)g(x) f(x)g(x) 3.6.14. f(x) Z[x] Z[x] Q[x]. f(x) 0 f(x) Z[x] Q[x] f(x) = g (x)h (x), deg g (x) 1, deg h (x) 1 g (x), h (x) Q[x] Q Z af(x) = g(x)h(x) a Z g(x), h(x) Z[x] a a a = 1 f(x) g(x) = αg 0 (x), h(x) = βh 0 (x), α, β Z g 0 (x) h 0 (x) a 1 p a p p af(x) = g(x)h(x) = αβg 0 (x)h 0 (x) g 0 (x)h 0 (x) p αβ p p α p β p α ( ) a α p f(x) = p g 0(x) (βh 0 (x)) Z[x] a

42 CHAPTER 3. 3.6.15 ( (Eisenstein) ). p f(x) = a 0 + a 1 x + + a n 1 x n 1 + x n Z[x] p a n 1, p a n 2,, p a 1, p a 0, p 2 a 0 f(x) Q[x]. f(x) Z[x] f(x) = g(x)h(x), g(x) = m i=0 b ix i Z[x], h(x) = l j=0 c jx j Z[x] a 0 = b 0 c 0 p b 0 c 0 p b 0, p c 0 p b 0, p b 1,, p b i 1, p b i i 0 i m < n f(x) x i b 0 c i + b 1 c i 1 + + b i 1 c 1 + b i c 0 p 3.6.16. p n 1 x n p n 1 n 3.6.17. (1) F p n N F = p n (2) p n N F = p n F F p p n p Z/pZ Z/pZ n f(x) (Z/pZ)[x]/f(x)(Z/pZ)[x] 3.6.18. p = 2 f(x) = x 2 + x + 1 (Z/2Z)[x] f(x) 1 Z/2Z f(x) 0, 1 0 f(x) (Z/2Z)[x] F = (Z/2Z)[x]/f(x)(Z/2Z)[x] 4 = 2 2 x α α 2 + α + 1 = 0 α 2 = α + 1 F = {0, 1, α, α + 1} + 0 1 α α + 1 0 0 1 α α + 1 1 1 0 α + 1 α α α α + 1 0 1 α + 1 α + 1 α 1 0 0 1 α α + 1 0 0 0 0 0 1 0 1 α α + 1 α 0 α α + 1 1 α + 1 0 α + 1 1 α

[1],, [2],, [3],, 43