A , . , .
1 ( ).
(+ mod 5)
{0, 1, 2, 3, 4}
(+ mod 5) | |||||
, , . , .
3 ( ). <A; W F; W R> , : A, W F, A, W R, A. A . , , , .
( ) .
, . : <A;f 1 ,...,fk;r 1 ,...,rl>, { f 1 ,...,fk } = W F, { r 1 ,...,rl } = W R.
4 ( ). <A;f 1 ,...,fk;r 1 ,...,rl> (n (f 1) ,...,n (fk)) (n (r 1) ,...,n (rl)), . <n (f 1) ,...,n (fk); n (r 1) ,...,n (rl)>.
2 ( ).
< N ;+,;<> <2, 2; 2>, +, . < N ;+,-;<> , -, .
G,
◦,
,
a∈G
e ◦ = ◦ = ,
a 1 ,
◦ a 1
= a 1◦ =e.
.
a 1 .
,
, ,
. ◦
, ,
.
,
. ,
|
|
◦
,
.
:
(a◦a)◦b = a◦b = (a+b)/2,
a◦(a◦b) = a◦((a+b)/2) = (3a+b)/4. :
Z , Q , R
;
Q\{0}, R\{0} \{0}
,
.
:
.
,
. ,
,
.
+,
x, (
, , ).
0 .
, ,
.
1 .
a 1
.
◦
( (( ◦ ) ◦ ) ◦ ) ◦ ,
◦ ◦ ◦ ◦.
k-
. k- k a,
k
. ,
0 a = 0 a (0)
= 1.
a b
a b= + (b)
a b
b.
a/b
a/b = b- 1
.
/b,
a b.
R
, R
:
( b) = a (b ),
a (b + c)=a b + a c,
(b + c) a = b a + c a.
0 = 0 = 0.
Z , Q
R
. ,
a b = 0 , = 0 b = 0,
. ,
.
.
' , ' = 1,
a. ,
- 1
.
. ,
, . ,
,
R,
|
|
R*.
F ,
.
.
Q
, R
.
F Q,
,
, Fᅫ Q. ,
, ,
.
. Q
.
.
, ,
.
.
q GF(q), Fq
.
.
GF(2).
, ,
2, .+ A A
B 0 1 b 0 1
0 0 1 0 0 0
1 1 0 1 0 1
. p , {0,
1, 2, , p 1} G(p).
(, , , )
p.
. p , n ,
, N = pn
,
N. , p = 2 n = 2
N = pn
= = 22
= 4.
4 2 ,
2 2=0 mod 4. . . ,
, G(N),
pn
= N . ,
pn
,
(n 1)
G(p).
, pn
G(pn
) ( G(N)). G(pn
)
.
q GF(q)*