Các số học trên vành các số nguyên gauss
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (317.84 KB, 38 trang )
!
"#$%&'()* +,-./"0
12345+627/89:
;<=2#!>9009?!@=
8"''=52?/"A450<<+
'B?2,-./C!@'<D/
E8:E4+,-./
++62"094%F%"+4E?2
2 G+GHGG+EIJ++K"LMN"MOPQ"PRMS
2?;ELT;U#$VU' /W@+
9 /EX /U'L" !MQ".M.N"@NYMM
L8E!@F#Z[
+62%'%?BZ\E
()*L8<E#Z'<D/E
()*"+44,]&Z2E./"!
^8"29_2D%B+()*
'8:E+'
6A+E,]&`G+aE<#Zb
62c+Gb2!/
I2G+++8#3#+!de+fghi2D%6F8#
MG9j"258T#$8#8-aE<#Zk@MlSN
VdG)O*W`G+#3#+`G+>92+
#_+G+GGNQMN@MljY"b6?]&
9%A#$/g,]&kE`G+
L39:",]&`G+a<#Z4
5#$\+,]&NNN#$8T-#[
g
!"#$%&
'%()!*$%&'%+)!,
d+m+/Z#8n%/Rc. .LT
oR>!6#8np2"0q5!
@
dZ9+,-C&?
"K /"K,-.-/r +#3,-/Ur>8-
#8n09+/!
dZ +#3,-/.A>0q"-%'!
$h/09+/!E#Z+6-
-/
sZZB9E9"6"2-=>Bm0q
+3/!tB
N N
- + =
>7+B+6/!"?!@"7
7nA4?4"+D#$4&"p2E9
2-F'
./$0(120(2
!"#$%!&
'()*+,
'''"-+./012/++' 7-]u!$
[
"
+4+#3"
;0 &
+f"45E,4u"e
mBG+"/4
•
• '
'''/342*56+.78' U
16/"?258T2f5+1v/"h
) * w b " x = + ∈ ⊂¢ ¢ £
£
N
M = −
) *¢
£
" ) *"
α β
∈¢
" b
α β α β
+ − ∈¢
[ ]
αβ
∈¢
-3 +=
W"V -
54/69Ey%ửứ ủieồm M
258T4aù ủoọ.
z{+"!
a]8#eD$[?6L+"?#9E
9#3<+]'E4":5
%9m"69E4%9A
="v-;<6!"#%ieọm|?
%'9"5!'Z%'/E
G]u"|E26#ZE9
E4L4%"?6
4 8 e
D$em"|}!U6!"4+f
|4<Dm[
''9':;<64%$5
6'78'9"#":$56;)
<(=
<2=
[ ]
Â
[ ]
Â
NM =+
[ ]
= + Â
WV
NN
. +==
[ ]
[ " V W Y Y. . = =Â Ơ
V W V W V W. . .
=
+=+=
"
V W V W V W. . .
=
V W V W V W. . .
=
WbWVVWVWV
NNNNNN
++=++
N N N N N N
V W V W V WV W + + = + +
''=/2)2>6?7@6A(BC-
7@6ADE6(F/06G1C-+./012/++
'''=/2)2>66G1C-
+./012/++'U"4+f#E
2E"%<'"?F-K'\+E%'
?BG+
#$/ %]?#EM"4
%"?Y]E+};0
&+f"8M#E/"_%]?
p?4#E/ !_%]E#$
#
''')HI'>?:$#@ ABC'9
6 !!"_
%]]E4 4~8#Z
4L#$-"?6]E;•!de+ft%eG
1"4%1
α
β
α
β
β
α
α β
) *
γ
∈¢
βαγ
=
¢
≠
£
[ ]
∗
¢
[ ]
¢
[ ]
¢
[ ]
∗
¢
{ } { }
M" ) *b V W M .
α α
= ± ± = ∈ =¢
α
M
) *
α
−
∈¢
MWMVWVWV
M
==
−
αα
M
V W" V W. .
α α
−
MWVWV
M
==
−
αα
V W M.
α
=
V W M.
α αα
= =
α
α
MWV
NN
=+=+ .
WY"MVW"V ±=
WM"YVW"V ±=
M±=+
±
;0&+f"t]
u}6-#$%?B+"%98•9:|
4" AD "%<'"?
L!de+f"%?%p%0%
Bem_%]
''9'J@6ADE6(F/042C-+./012/++
K'B+/D+B'?_2D%B"%
'4+A#Z#+
4 #@EAB;
+?%9%]?61%]V%4W1%
V%4WL4%"2D%B?4%Y
%94#C#&+f"?2D%B+6
9#3 "#$-%90"v-N%9
_2D%BE2\6 €_2D%B
?Z
'9'KL/MNC-BMOD42.P
42C-+./012/++
<DE|ed]e^8#
M
) * w ) *b ) *x
α α
∗ −
= ∈ ∈¢ ¢ ¢
α β
:
βα αβ
) *
α
∈¢
) *¢
αα
"Y≠
βγα
=
β
γ α
:
γ
β α
:
∈¢
) *¢
) *¢
N
WMVN +=
'9'')HI'<FGH"
'9$I="9$I'%&J
E$I'9
6,1 +1
v +: +:
"!<!54/ /
5/DK4"%=%9#$B
_8#3e69Z]"u"4+9•Z
M,4u‚M~4
Lmde+f"1#+' 4+8Dƒ
"KNQ' /'
+ 4
.
D 45D#ZEe^8"%
4!#$8#,ve^8
"KNQ';[„…Sl†NjN"„QY†QY 4[
α
β
Y≠
β
ρµ
"
ρβµα
+=
WVWV
βρ
<
- +=
β
α
"- ∈¡
£
¡
¡
) *
µ
∈¢
- +
µ
- +
N
N
≤
WV
µ
β
α
−.
WVWVWVWWVVWV
µµ
β
α
βµ
β
α
ββµα
〈−=−=−
γβ
"
S‡"PQ +=−=
βα
QP
‡M
QP
MM
WS‡WVS‡V
WS‡WVPQV
S‡
PQ
−=
−+
−−
=
+
−
QP
‡M
QP
MM
−
−
+=+=
βα
"
−=−= N
βµαρ
P Q V‡ S WV W VN W" − = + − + −
QP −
S‡ +
MYS MSO
QY QY
= +
d]; #ZD
+6€8#m"+1 v"#Z #$25]
2fDG"f+99 8Z]jpj
"99#$9•m-~%=5;%9
BM"+]E;p4525]2\jp €j
#3+A2%<Z]!jp+mV#3+A9
d-WL?5f+#3+A6;45!+]-m
#3+A4L66€D;pf+S#3+A4m5
9•"8445!++]2f[N†NbN†SbS†S
K?BD
F - E !
^8 +
!
8: 1
e6#$#DEGu[
VMWb
VNWU/6
1%?1
ˆDE
VF-#$+\+W%98DL? #
" ) *" Y
α β β
∈ ≠¢
δ
"
α β
βδαδ
"
δ
′
βδαδ
′′
"
δ
′
δ
WVWV
δδ
<
′
δ
βα
"
δ δ
′
:
w b x
δ δ δ
′ ′
:
"
α β
" ) *
µ γ
∈¢
'
µα β δ
+ =
DE6/}#D!!#
DEXZ?="!^8}_
,v#$2??8#/vJGy
4+f%Y•?M#
DE0
: ;<6 K L
'9M'9A'9LA
NM'9$I
<^8E +#3$12'EJa
'9''RSHI'"9$ I'%EAB;.DO
6 _ +f
%9#EK4"_8:!^82V%'2D%B
2=F-#DE•
U62D%BG?1_%]1
_%?L#%95%?E6%44
"8nL#!_%] +4tv
%eG 4ƒ
L#'B"F-e ^8+"
4,]&Z2E./#[
Y"Y ≠≠
) *¢
γβα
""
α
βγα
βα
γα
α
β
αβ
"
α
Y≠
α
δ
αβ
"
'"
µ
'
αβµδ
+=
αδ
α
δ
α
δ
α
α δδ β
α β
δ
'
γαδγβµδγ
MM −−
+=
βγα
γα
) *¢
'9'9'BMO V,]&Z2E./WPQ
IA$07R:'D%J
"
"9 # @ AB
C"9$#@EAB;<ASEDT?UASE
DSAS"AD=$TU9"9E)VW.D"9
TUXY'9&J$'C?Z(2[\
'%Q
6.F-[ ?
B-G +#3$#36%4_%]
_m<#!F-/8#Z
4 L? _2D%B6]&
#$_%B"?_%9%]~
I8:?B-!#$m<•>
E]&
<8D[L?_
%]6,]&5_%9%]K9D<a
B"45m<<_2D%B8#8-
_2D%B"%9D?9%9%?;0&+f"_
%]Em<#$!+_2D%B
M
α εγ γ
=
ε
γγ
""
M
δδα
M
∈
′
=
α
δ
V W
δ
δ γ
:
WV
α
.
MWV =
α
.
α
[ ]
β
∈¢
A. <WV
β
M>A
α
A. =WV
α
αα
'
µα
=
'"
µ
WVWVWV '
µα
=
WVW"VM A' <<
µ
µ
'
αα
γγγα
NM
=
γ
_4m<%
‰8:JaMSN\+"tv
+%?+
_4 !!"8"F- ?#}
_2D%B4Jm3"?Ev
!#$v#Z8Em<\h?M ?
?#!!#$ƒ
:;<6']C"^B_Q '9S6RU
%"%E'9`E<9CV99$
Sa=>XDTaASB"9$
ERJ":9
\+#$JaMSS
'9'T')HI'"9$ I.D
M
'T'U67@6ADE6(F/042C-2/++
,]&Z2E./ >?!\+DD
?E'_%]_2D%BE
,_2D%B"+#?4%?B
δδα
M
=
δδγγ
MM
=
M
γ
γ
δδγ
MM
δγ
M
δ
M
γ δ
:
δδγγ
MM
=
M
γ
) *¢
" "
α β γ
α βγ
"
α β
α γ
[ ]
¢
'T'')HI'"9#@
EAB;&JE+
6 4L#!"G
JaMSN"#E#4E.G#!
8D !!"?F- G,]&JGy
"F-b;Y("84"mn
?2D%Bƒ
: ;<6. P 7 B ETc Q#
@EAB;'97 9]T"9SB
#$#@EAB;'9
G ' MjM ,] & MSS"'_2D%
BE#Z#Z'm_2D%BE
9#3+ 27„N
'T'9'RSHI'IB@"9
>D"9#@EAB;
6 !!"?6
"+1u+_%]ƒ
'T'T')HI'$%EAB; 2'9 "9'9$#@
"AD'%"9$
6 4}
#_%?"u1%2D%B6
[ ]
γ
∈¢
∈¢
γ
[ ]
¢
V W.
γ γ γ γ
=
γ
( )
.
γ
; ≠
;
γ
M
γ
γ
[ ]
¢
[ ]
¢
[ ]
¢
[ ]
¢
[ ]
α
∈¢
V W.
α
α
α βγ
=
V W V W V W. . .
β γ α
=
V W M.
β
=
V W M.
γ
=
"
β γ
[ ]
¢
M +
M ± ±
N VM WVM W = + −
M N+
M +
M" ± ±
M +
VM W N. + =
GJaMjS"4ƒ
Ui" +#3$[
V8jWV8jW
'T'V' ) HI ' .D "9 <!="9
#@EAB;
6_ %92D
%B ?_%9%]"#! t<mE
|
8"+U?"?6
V8jW"#F8#+{+%95d|+L#!_2D
%BEƒ
'T'W' B MO' IB @ "9 '9<8jWW
<(= >& J E
<=U-$%6Xb
<2= $%EAB;
&'9$#@"AD'%N
,5/"
„Q45?8D8#8-aE<#Z.Q
M ≡
S≡
S≡
[ ]
¢
αβ
=
"
α β
V W" V W M. .
α β
>
N
V W V W V W. . .
α β
= =
V W" V W M. .
α β
>
V W V W. .
α β
= =
" "
α
= + ∈¢
N N
V W S.
α
= + = ≡
) *¢
α
N N
+ =
) *¢
" + −
N N
+ =
N N
Q M N= +
) *¢
M N "M N + −
M N+
{ }
M N " M N " N "N + − − − + −
M N−
{ }
M N " M N "N " N − − + + − −
%92D%B+4#2D%B;4uQ4P#2D%
BF_%?"4_%?"4
,]&Mjl2f'7-2a
'T'X'RSHI <d)= "9
<!=&J
6 !!"1 G
,]&Š"4
V8W
1%"4[
tF8#+EJaC+Gƒ
G Ja C+G" F -
L#!"
?de+"4 %9#Ebe
V6W t4"+%9_2D%B ?
_%9%]"#! t<mE|
M≡
N
M +
j M A= +
Vj W‹ MA ≡
N N N
Vj ‹W VMNN WVVN MWVN NWj WW
VMNN WVV N WV N MWV MWW V MW VMNN W VN ‹W V8 W
A
A A A A A
A A A A A
= + +
≡ − − + − ≡ − ≡
N
M +
V WV W + −
M
) *
± ∉¢
αβ
=
"
α β
V W" V W M. .
α β
>
N
V W V W V W. . .
α β
= =
8"+U?"?6
Kv]VMWE,]&Mjl#$
/ #E +{+
$}#EVpD$?Em<E
<_2D%B+W €p+#2D%B8
DV%e%?WE !!_V84W#2D%
BE%9%? +
V+W
,8n?V+"
#Z#Z+W 1mnU6!"%v]VNW
E,]&Mjl#$ƒ
Kv]VMWE,]&Mjl#$2??8#/,]&`G+
K?$'Mjj"MjQ,]&Mjl#$%?B
'T'Y'BMO'$#@EAB; &
<(='9$#@"AD V"9f
<2=$<8
jW'9$#@"ADV"9f
<+=g#@EAB
;TUU$T@EAB;'9$
V W" V W M. .
α β
>
V W V W. .
α β
= =
" "
α
= + ∈¢
N N
V W.
α
= + =
) * + ∈¢
−
[ ]
¢
" + −
+ −
" + −
V WV WV WV W + − + −
[ ]
¢
N N N N
V WV W + +
[ ]
¢
¢
[ ]
¢
M +
M ± ±
S ≡
{ }
" ± ±
" + −
MV8jW ≡
N N
+ =
#@"AD$<=R:O"9
EU-$%E'9%6X`h
CH
Z[\]&"#$%!&
''BMO^_G*26CI6S2+.K7
'''BMO^_G*26CI6S422+.K7'."iR
j:%Jk<#ZW„dN†N"d"
ZY"%p%F8#MG9j
"KNQ;iQ"MS"M‡"NO"S‡"jMF8#MG9Rj"8
40258T#$8#8-aE<#Z[
Q„M
N
†N
N
bMS„N
N
†S
N
bM‡„M
N
†j
N
bNO„M
N
†Q
N
bS‡„M
N
†l
N
1%"i‡"MM"MO"NSSMF8#SG
9j"840%95258T#$8#8-aE<
#Z
I2G+++8#3#+!de+fghi2D%6F8#
MG9j"258T#$8#8-aE<#Zk@MlSN
VdG)O*W`G+#3#+`G+>92+
#_+G+GGNQMN@MljY"b6?]&
9%A#$/]&E`G+
'''D`8*42BMOCI6S422+.K7
L?i258T#$8#8-aE<
#Z"68<#Z%jp8#Y1M"jp
458#MU6!"%'E]&5UDA-
%'E.m09+62E^G+
^G+ >9g]C"^l)'7kU
k@M‡j‡"%>jYaŒ92+#_
82lQ@M‡j‡;F4Q2#b2#
@#$+62+#_82@M‡jO +
+628#m"2#M"N"S8E^G+"2#
jQ4_aV8•,]&C+GW
NW >U9m"9kUn"9k
UW
;8,]J+r`2[
V
N
†
N
WV
N
†
N
W„V†W
N
†V…W
N
2.DX9D '9B"o7R
Rj9kU nRRj9
kUW
+#?258T[
„
N
†
N
b„
N
†
N
"
J\6[
V†WV…W„
N
N
…
N
N
„
N
V
N
†
N
W…V
N
†
N
W
N
„
N
…
N
?""+V†28W1Vr28W?L?V
r28W?"6_8:]J+R`2"4[
8 < e =
?"<#Z""}84
<#Z.+258T#$aE<#Z
+#3$A-<b=?"0m<[
1-2#
#Z#+
+.D D 9R Rj9kU
pASq %ASRRj9k
UW
;2f_ /#E45258T
aE<#Z,1[
V%9
D?9%W
~258T#$a
E<#Z"8:2#N?%+[
N
N
V W
−
N
N
V W
+
M N A
= L
45258TaE<#Z 1mn
!.D'9MQ%
2
b
2
7RRj
9kUW
;_F-"•
N
†
N
4<D#%95258TaE<
#Z +14de1<=•>a<b=•Dse-
#E
N
†
N
%95258TaEN<#Z,1[
+4"ZY%9#$B-e(
4[
N
†
N
„
N
-
N
†N-†
N
†
N
-
N
†N-†
N
„/-†V
N
†
N
W
.+
N
†
N
?-L?•68ab‚b
nn?a<b=•DU!#D
E2fZM
L?-%9•"6F-
-•?"+"}?Vmn?
•WU!-•L?1[
"
M N A
=
L
6
M
"
M
• ?-"+{+
M
†
M
‚†"
nn?a<b=•D 4
rPQ"iJ!Ab(7Rj9kU
W
G,]&`G+2e"mF8#MG9[
se'=?[
jA
…V…MW
jA
„V
NA
†V…MW
NA
WV
NA
…V…MW
NA
W"
-tN?j%
~"<D+
V
NA
†V…MW
NA
W"V
NA
…V…MW
NA
W
?L?F-V
NA
†V…MW
NA
W?"6G2#j"+
45258TaE<#Z
L#$-"_%9F-5V
NA
†V…MW
NA
W?"+[
V
NA
…V…MW
NA
W
?/-tN?!AL#!F8#9
G9[
~4"#Z+6F8#[
N N
M M
+
4j%'"mn,]&C+G#$258#mU!
_E0.+45258T8#8-aE<
#Zƒ
NMS,]&C+G ;
HVdW'"42!L?4<D'EHVdW
%9?"6#Z+6F8#4%9B'++#3
8
; B-G
,]&50„Y
_]&0„%"de
%9FD%9
8GVHW„%†M"'
K9D<aB_•Y"!F-+HV+W„YK4[
42!•%†Mz{+"VdW%9FD%9"84VdW4%9B%'
K?$Vd…+W40'"+HVdW4%9B%†M'.+
ƒ
''(>6F/(;/F/2
BMO^_G*26CI6S422+.K7
NNM;8EC+G
( )
V W Y 8s - ≡
¢
`G+>92N%?B%+#_JGcNQ
O@MlQj[
•
•
Œ}?[
t.D9?M"9+Ou'9"%+!
9T'%UN kU'9r"#
UA$,
L4%"?;48-NYA†S1NYA†‡"6;„-
N
†Q
N
.
"^G+>\+8[
•
•
,]&`G+a<#Z45#$\+#[
'''BMO'
!"#$
%&'%()!*$%&'%+)!
6 G,]&Z2E./+4
5?
"
N N
- + =
M M
M M M M
N
M M M M M
N V W VM W
A A
A A
γ γ γ γ γ γ γ γ
= = − +
+4_2D%B E"F8#
SG8j48-F8#MG8j
_ G<8D
Em<+<_2D%B+"+_%?
E
VMW
L#!'E
#Z+62f[
• Y?+ib
• ?DŽ
;0&+fm_jdD'\
9+8'258T\+%2\+2_2D%
]
1%"/#iE 4 5?8#8-<_2D
%B+
"
L#V8jW/
V8jW/V8jWV8NW
#Z#Z8n?'E # V8jW# V8jW
E2f
"
v
γ
) *¢
=
v
γ
+
N N
+
N N
V WV W
- - - = + = + −
) *¢
- +
M
M M
MN N
M M M
VM W
A
A
γ γ γ γ
− −
+
Y
v v
≤ ≤
N N
- + =
M
jV MWV MW
+ +
- +
{ }
M"
± ±
[ ]
¢
M M M
M M M
A
' '
' '
A
γ γ γ γ
−
−
=
Y "Y
v v
' ≤ ≤ ≤ ≤
S
≡
M
v v
γ γ
≡
v
M ≡
M
M
A
⇔ + + ≡
M≡
S≡
• Y?++ib
• ?DŽ
U!"4ƒ
'9'ZNQ+.P42C-+./012/++
'9'')HI'F
E
6_
E#Z+6>L?Ž6V8PW#RM%9<#Z
9PU!iŽU?-#Z+68#8-
+#?4
'9''RSHI'4%Q$6'9 "9M
6 ! !"
_"#!4+41%L#!8#Zt
#Ejt#Ei84_%]ƒ
2?+f,]&Z2E
./A0!_%]E1%"G2a
M
V MWV MW
+ +
N S
M -= −
V " W VM"YW- =
V " W-
-
N
M ≡ −
-
S
V WV W- = + −
N ∈ ¢
+ −
) *¢
γ
∈¢
"
γ γ
+ −
N
γ
V W VN W j. .
γ
=
N
V W V W M . .
γ γ
+ ⇒ + = +
V W.
γ
N
M +
V W M.
γ
=
γ
) *¢) *¢
{ }
M"
± ±
" + −" + −
) *¢
S
V W + = +
S
V W − = −
S
V W + = +
N N
M VS W = −
V " W VY" MW = −
Y =
M- =
+"•L#!!#ZE_EV8/
_%]!#ZE_%]4W~4F
-V2fD$4W.m
E?Ev#$#Z+6'Z
#$ tm"+#!ƒ
J/+%6X/•oNYMY45#$25
%;8•%98:+?E"
52?":5\mv8<mE
|"45•Z+62m
'9'9')HI'4%Q
NAST
6JBm?'%9m"5/
?''%9m +#?!de+fYM"N
2#3
m'
E#Z+6>-['Ž'iGu9
#3L+"%&' #Z+6 €+f" 40
N N
MQ j
- + =
N N
N MQY jb+ =
N N N N N
j MQY M MQM j+ = + =
V WF
N N
MQ j
- + =
V WF
M −
