(K,+, ) . (K, +) ,
- 1. (K,+, ) 0 a ∈ K −a.
- 2. ∀ a, b, c ∈ K (a + b = a + c ⇒ b = c).
- 3. a, b ∈ K K a − b, a − b = a + (−b). , K , 1′8′.
- 4. K , .. ∀ a, b, c ∈ K ((a − b)c = ac − bc ∧ c(a − b) = ca − cb).
-. a, b, c ∈ K. K + , (a − b)c + bc = ((a − b) + b)c = ac, , (a − b)c = ac − bc.
.
- 5. ∀ a ∈ K a0 = 0a = 0.
. a ∈ K b K. b − b = 0 , , a0 = a(b − b) = ab − ab = 0.
, 0a = 0.
- 6. ∀ a, b ∈ K (−a)b = a(−b) = −(ab).
. a, b ∈ K. (−a)b + ab = ((−a) + a)b =
= 0b = 0. , (−a)b = −(ab).
a(−b) = −(ab).
- 7. ∀ a, b ∈ K (−a)(−b) = ab.
. , , (−a)(−b) = −(a(−b)) = −(−(ab)) = ab.
. 6 7 .
K 6 7
- 8. k, l . ∀ a, b ∈ K (ka)(lb) = (kl)ab.
(K,+, ) H K, + , K, .
:
, Z (Q,+, ), Q (R,+, ), Rn×n (Cn×n,+, ), Z[x] (R[x],+, ), D[a,b] (C[a,b],+, ).
(K,+, ) K, {0} (K,+, ). (K,+, ).
.
H (K,+, ), .. (H,+, ) . , (H, +), .. H (K, +). .
- 1. H K K.
|
|
- 2. a H K H −a, .. K.
- 3. a b H H a − b, .. K.
.
1 ( ).
H K + K , :
∀ a, b ∈ H a + b ∈ H, (1)
∀ a ∈ H − a ∈ H, (2)
∀ a, b ∈ H ab ∈ H. (3)
-.
. H (K,+, ). H (K, +). ( ), H (1) (2). , H , K, .. H
(3).
. H ⊂ K, H 6= ∅ H (1) − (3). (1) (2) , H (K, +), .. (H, +). , (K, +) , (H, +) . , (3) , H. H + , + K.
2 ( ).
H K +
K . . , :
∀ a, b ∈ H a − b ∈ H, (4)
∀ a, b ∈ H ab ∈ H. (5)
1.
2′ ( ) .
7. (, , , ).
e 0, , .
(Q,+, ), (R,+, ), (C,+, ).
1. F
, , ..
∀ a, b, c ∈ F (ab = ac ∧ a 0 ⇒ b = c).
2. F .
3. (K,+, )
, K \ {0} .
4. (K,+, ) .
.
(F,+, ).
a b F, b 0,
c ∈ F, a = bc.
1. a b F, b 0, a/b, a/b= ab−1.
2. ∀ a ∈ F \ {0}
a/a= e ∀ a ∈ F a/e= a.
3. ∀ a, c ∈ F ∀ b, d ∈ F \ {0}
a/b=c/d ⇔ ad = bc.
4. ∀ a, c ∈ F ∀ b, d ∈ F \ {0}
a/b*c/d=ac/bd
5. ∀ a ∈ F ∀ b, c, d ∈ F \ {0}
(a/b)/(c/d)=ad/bc
6. ∀ a ∈ F ∀ b, c ∈ F \ {0}
|
|
ac/bc=a/b
7. ∀ a ∈ F ∀ b, c ∈ F \ {0}
(a/b)/c=a/bc
8. ∀ a, b ∈ F ∀ c ∈ F \ {0}
ab/c=ab/c
F, p (F, +), p.
F , (F, +), 0.
8. (, , , )
(F,+, ) S F, + , F, .
Q (R,+, );
R (C,+, );
.
1. S F
F.
2. a S F S −a, .. F.
3. a b S F
S a−b .. F.
4. S F
e F.
5. a S F, -
, S a−1, .. , a F.
.
1 ( ).
H F c +, ,
, (F,+, ) , :
∀ a, b ∈ H a + b ∈ H, (1)
∀ a ∈ H − a ∈ H, (2)
∀ a, b ∈ H ab ∈ H, (3)
∀ a ∈ H \ {0} a−1 ∈ H. (4)
2 ( ).
H F c +, ,
, (F,+, ) , :
∀ a, b ∈ H a − b ∈ H, (5)
∀ a ∈ H ∀ b ∈ H\{0} a/b ∈ H. (6)
10. Z
: a,b,c R, :
1) |b, b|c => a|c
2) a|b, a|c => a| (b c)
3) a|b => a|bc
a, b Z :
1) a|b ó ≠ b
2) a|b, b≠0 => |a|≤|b|
3)a|b b|a ó |a|=|b|
b, q r, a=b*q + r, 0≤r≥|b|, q , r-
: a b Z, b≠0, b , .
, a b Z, b≠0, b|a ó
11.
() Z d,
1) d .. d| , d| d|
2) d .. d| , d| d| => d| , d| d|
: ,
=0, , , .
: n≥2 , ( =((() ), )
() ,
1. ,
2. K
3.
: n≥2 , =0, ≠0 , , .
a b [a,b]=|a*b|/(a,b)
: n≥2 ,
|
|
[ ]=[[