Ring of integers modulo 23\newcommand{\Bold}[1]{\mathbf{#1}}\ZZ/23\ZZ Ring of integers modulo 23\newcommand{\Bold}[1]{\mathbf{#1}}\ZZ/23\ZZ |
Finite Field of size 23\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{F}_{23} Finite Field of size 23\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{F}_{23} |
Ring of integers modulo 23\newcommand{\Bold}[1]{\mathbf{#1}}\ZZ/23\ZZ Ring of integers modulo 23\newcommand{\Bold}[1]{\mathbf{#1}}\ZZ/23\ZZ |
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\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True} |
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\newcommand{\Bold}[1]{\mathbf{#1}}6
\newcommand{\Bold}[1]{\mathbf{#1}}6
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+ a b c d e f g h i j k l m n o p q r s t u v w
+----------------------------------------------
a| a b c d e f g h i j k l m n o p q r s t u v w
b| b c d e f g h i j k l m n o p q r s t u v w a
c| c d e f g h i j k l m n o p q r s t u v w a b
d| d e f g h i j k l m n o p q r s t u v w a b c
e| e f g h i j k l m n o p q r s t u v w a b c d
f| f g h i j k l m n o p q r s t u v w a b c d e
g| g h i j k l m n o p q r s t u v w a b c d e f
h| h i j k l m n o p q r s t u v w a b c d e f g
i| i j k l m n o p q r s t u v w a b c d e f g h
j| j k l m n o p q r s t u v w a b c d e f g h i
k| k l m n o p q r s t u v w a b c d e f g h i j
l| l m n o p q r s t u v w a b c d e f g h i j k
m| m n o p q r s t u v w a b c d e f g h i j k l
n| n o p q r s t u v w a b c d e f g h i j k l m
o| o p q r s t u v w a b c d e f g h i j k l m n
p| p q r s t u v w a b c d e f g h i j k l m n o
q| q r s t u v w a b c d e f g h i j k l m n o p
r| r s t u v w a b c d e f g h i j k l m n o p q
s| s t u v w a b c d e f g h i j k l m n o p q r
t| t u v w a b c d e f g h i j k l m n o p q r s
u| u v w a b c d e f g h i j k l m n o p q r s t
v| v w a b c d e f g h i j k l m n o p q r s t u
w| w a b c d e f g h i j k l m n o p q r s t u v
\newcommand{\Bold}[1]{\mathbf{#1}}
+ a b c d e f g h i j k l m n o p q r s t u v w
+----------------------------------------------
a| a b c d e f g h i j k l m n o p q r s t u v w
b| b c d e f g h i j k l m n o p q r s t u v w a
c| c d e f g h i j k l m n o p q r s t u v w a b
d| d e f g h i j k l m n o p q r s t u v w a b c
e| e f g h i j k l m n o p q r s t u v w a b c d
f| f g h i j k l m n o p q r s t u v w a b c d e
g| g h i j k l m n o p q r s t u v w a b c d e f
h| h i j k l m n o p q r s t u v w a b c d e f g
i| i j k l m n o p q r s t u v w a b c d e f g h
j| j k l m n o p q r s t u v w a b c d e f g h i
k| k l m n o p q r s t u v w a b c d e f g h i j
l| l m n o p q r s t u v w a b c d e f g h i j k
m| m n o p q r s t u v w a b c d e f g h i j k l
n| n o p q r s t u v w a b c d e f g h i j k l m
o| o p q r s t u v w a b c d e f g h i j k l m n
p| p q r s t u v w a b c d e f g h i j k l m n o
q| q r s t u v w a b c d e f g h i j k l m n o p
r| r s t u v w a b c d e f g h i j k l m n o p q
s| s t u v w a b c d e f g h i j k l m n o p q r
t| t u v w a b c d e f g h i j k l m n o p q r s
u| u v w a b c d e f g h i j k l m n o p q r s t
v| v w a b c d e f g h i j k l m n o p q r s t u
w| w a b c d e f g h i j k l m n o p q r s t u v
\newcommand{\Bold}[1]{\mathbf{#1}} |
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* a b c d e f g h i j k l m n o p q r s t u v w
+----------------------------------------------
a| a a a a a a a a a a a a a a a a a a a a a a a
b| a b c d e f g h i j k l m n o p q r s t u v w
c| a c e g i k m o q s u w b d f h j l n p r t v
d| a d g j m p s v b e h k n q t w c f i l o r u
e| a e i m q u b f j n r v c g k o s w d h l p t
f| a f k p u c h m r w e j o t b g l q v d i n s
g| a g m s b h n t c i o u d j p v e k q w f l r
h| a h o v f m t d k r b i p w g n u e l s c j q
i| a i q b j r c k s d l t e m u f n v g o w h p
j| a j s e n w i r d m v h q c l u g p b k t f o
k| a k u h r e o b l v i s f p c m w j t g q d n
l| a l w k v j u i t h s g r f q e p d o c n b m
m| a m b n c o d p e q f r g s h t i u j v k w l
n| a n d q g t j w m c p f s i v l b o e r h u k
o| a o f t k b p g u l c q h v m d r i w n e s j
p| a p h w o g v n f u m e t l d s k c r j b q i
q| a q j c s l e u n g w p i b r k d t m f v o h
r| a r l f w q k e v p j d u o i c t n h b s m g
s| a s n i d v q l g b t o j e w r m h c u p k f
t| a t p l h d w s o k g c v r n j f b u q m i e
u| a u r o l i f c w t q n k h e b v s p m j g d
v| a v t r p n l j h f d b w u s q o m k i g e c
w| a w v u t s r q p o n m l k j i h g f e d c b
\newcommand{\Bold}[1]{\mathbf{#1}}
* a b c d e f g h i j k l m n o p q r s t u v w
+----------------------------------------------
a| a a a a a a a a a a a a a a a a a a a a a a a
b| a b c d e f g h i j k l m n o p q r s t u v w
c| a c e g i k m o q s u w b d f h j l n p r t v
d| a d g j m p s v b e h k n q t w c f i l o r u
e| a e i m q u b f j n r v c g k o s w d h l p t
f| a f k p u c h m r w e j o t b g l q v d i n s
g| a g m s b h n t c i o u d j p v e k q w f l r
h| a h o v f m t d k r b i p w g n u e l s c j q
i| a i q b j r c k s d l t e m u f n v g o w h p
j| a j s e n w i r d m v h q c l u g p b k t f o
k| a k u h r e o b l v i s f p c m w j t g q d n
l| a l w k v j u i t h s g r f q e p d o c n b m
m| a m b n c o d p e q f r g s h t i u j v k w l
n| a n d q g t j w m c p f s i v l b o e r h u k
o| a o f t k b p g u l c q h v m d r i w n e s j
p| a p h w o g v n f u m e t l d s k c r j b q i
q| a q j c s l e u n g w p i b r k d t m f v o h
r| a r l f w q k e v p j d u o i c t n h b s m g
s| a s n i d v q l g b t o j e w r m h c u p k f
t| a t p l h d w s o k g c v r n j f b u q m i e
u| a u r o l i f c w t q n k h e b v s p m j g d
v| a v t r p n l j h f d b w u s q o m k i g e c
w| a w v u t s r q p o n m l k j i h g f e d c b
\newcommand{\Bold}[1]{\mathbf{#1}} |
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\newcommand{\Bold}[1]{\mathbf{#1}}5
\newcommand{\Bold}[1]{\mathbf{#1}}5
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Univariate Polynomial Ring in x over Ring of integers modulo 23\newcommand{\Bold}[1]{\mathbf{#1}}\ZZ/23\ZZ[x] Univariate Polynomial Ring in x over Ring of integers modulo 23\newcommand{\Bold}[1]{\mathbf{#1}}\ZZ/23\ZZ[x] |
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\newcommand{\Bold}[1]{\mathbf{#1}}x^{5} + 13 x^{4} + 5 x^{3} + 20 x^{2} + 22
\newcommand{\Bold}[1]{\mathbf{#1}}x^{5} + 13 x^{4} + 5 x^{3} + 20 x^{2} + 22
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\newcommand{\Bold}[1]{\mathbf{#1}}(x + 18) \cdot (x + 19) \cdot (x + 20) \cdot (x^{2} + 2 x + 5)
\newcommand{\Bold}[1]{\mathbf{#1}}(x + 18) \cdot (x + 19) \cdot (x + 20) \cdot (x^{2} + 2 x + 5)
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y^5 + 59*y^4 + 1201*y^3 + 9289*y^2 + 19090*y + 34200 y^5 + 59*y^4 + 1201*y^3 + 9289*y^2 + 19090*y + 34200 |
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\newcommand{\Bold}[1]{\mathbf{#1}}\left[22, 0, 20, 5, 13, 1\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[22, 0, 20, 5, 13, 1\right]
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Met polynomen kan je het algoritme van Euclides toepassen, je kan hun grootste gemene deler bepalen, en de polynomen $\mu(x)$ en $\lambda(x)$ waarvoor $\mu(x)a(x)+\lambda(x)b(x)=\text{ggd}(a(x),b(x))$.
a = 6*x^4 + 13*x^3 + 12*x^2 + 11*x + 20 b = 13*x^4 + 21*x^3 + 6*x^2 + 2*x + 8 ggd(a,b) = x + 16 a = 6*x^4 + 13*x^3 + 12*x^2 + 11*x + 20 b = 13*x^4 + 21*x^3 + 6*x^2 + 2*x + 8 ggd(a,b) = x + 16 |
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\newcommand{\Bold}[1]{\mathbf{#1}}\left(x + 16, 7 x^{2} + 22 x + 17, 18 x^{2} + 21 x + 17\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(x + 16, 7 x^{2} + 22 x + 17, 18 x^{2} + 21 x + 17\right)
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\newcommand{\Bold}[1]{\mathbf{#1}}x + 16
\newcommand{\Bold}[1]{\mathbf{#1}}x + 16
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Beschouw nu het quotiënt van de polynoomring $R = \mathbb{F}_{23}[x]$ modulo het polynoom $b(x)$.
Univariate Quotient Polynomial Ring in t over Ring of integers modulo 23 with modulus x^4 + 14*x^3 + 4*x^2 + 9*x + 13 13*x^4 + 21*x^3 + 6*x^2 + 2*x + 8 Univariate Quotient Polynomial Ring in t over Ring of integers modulo 23 with modulus x^4 + 14*x^3 + 4*x^2 + 9*x + 13 13*x^4 + 21*x^3 + 6*x^2 + 2*x + 8 |
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\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{False}
\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{False}
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\newcommand{\Bold}[1]{\mathbf{#1}}0
\newcommand{\Bold}[1]{\mathbf{#1}}0
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\newcommand{\Bold}[1]{\mathbf{#1}}3 t^{3} + 6 t^{2} + 4 t + 9
\newcommand{\Bold}[1]{\mathbf{#1}}3 t^{3} + 6 t^{2} + 4 t + 9
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\newcommand{\Bold}[1]{\mathbf{#1}}13 x^{4} + 21 x^{3} + 6 x^{2} + 2 x + 8
\newcommand{\Bold}[1]{\mathbf{#1}}0
\newcommand{\Bold}[1]{\mathbf{#1}}13 x^{4} + 21 x^{3} + 6 x^{2} + 2 x + 8
\newcommand{\Bold}[1]{\mathbf{#1}}0 |
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(13) * (x + 8) * (x + 16) * (x + 17) * (x + 19)
\newcommand{\Bold}[1]{\mathbf{#1}}
(13) * (x + 8) * (x + 16) * (x + 17) * (x + 19)
\newcommand{\Bold}[1]{\mathbf{#1}} |
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\newcommand{\Bold}[1]{\mathbf{#1}}0
\newcommand{\Bold}[1]{\mathbf{#1}}0
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Wanneer we een irreducibel polynoom uitdelen, dus doen alsof die 0 is, krijgen we een eindig veld.
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\newcommand{\Bold}[1]{\mathbf{#1}}\ZZ/23\ZZ[x]
\newcommand{\Bold}[1]{\mathbf{#1}}\ZZ/23\ZZ[x]
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We zoeken een irreducibel polynoom door er enkele te proberen...
14*x^4 + 11*x^3 + 9*x^2 + 11*x + 15\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True} \newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{False} 14*x^4 + 11*x^3 + 9*x^2 + 11*x + 15\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True} \newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{False} |
Univariate Quotient Polynomial Ring in x over Ring of integers modulo 23 with modulus x^4 + 9*x^3 + 22*x^2 + 9*x + 6 Univariate Quotient Polynomial Ring in x over Ring of integers modulo 23 with modulus x^4 + 9*x^3 + 22*x^2 + 9*x + 6 |
True False Ring of integers modulo 23 3404825447 = 23^7 Univariate Quotient Polynomial Ring in x over Ring of integers modulo 23 with modulus x^7 + 18*x^6 + 2*x^5 + 22*x^4 + 10*x^3 + x^2 + 5*x + 3 True False Ring of integers modulo 23 3404825447 = 23^7 Univariate Quotient Polynomial Ring in x over Ring of integers modulo 23 with modulus x^7 + 18*x^6 + 2*x^5 + 22*x^4 + 10*x^3 + x^2 + 5*x + 3 |
Hoewel dit de manier is waarop ze eigenlijk gemaakt worden, is de kortste manier om deze velden in Sage te definiëren, de volgende:
Finite Field in a of size 23^7 Finite Field of size 23\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{F}_{23^{7}} Finite Field in a of size 23^7 Finite Field of size 23\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{F}_{23^{7}} |
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\newcommand{\Bold}[1]{\mathbf{#1}}12 a^{7} + 9 a^{6} + a^{5} + 11 a^{4} + 5 a^{3} + 12 a^{2} + 14 a + 13
\newcommand{\Bold}[1]{\mathbf{#1}}12 a^{7} + 9 a^{6} + a^{5} + 11 a^{4} + 5 a^{3} + 12 a^{2} + 14 a + 13
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\newcommand{\Bold}[1]{\mathbf{#1}}a
\newcommand{\Bold}[1]{\mathbf{#1}}a
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\newcommand{\Bold}[1]{\mathbf{#1}}8 a^{5} + 10 a^{4} + 15 a^{3} + 4 a^{2} + 6 a + 10
\newcommand{\Bold}[1]{\mathbf{#1}}8 a^{5} + 10 a^{4} + 15 a^{3} + 4 a^{2} + 6 a + 10
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Nu iets kleiner, een eindig veld van orde 9. Als we geen irreduciebel polynoom opgeven, dan kiest Sage er zelf één.
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\newcommand{\Bold}[1]{\mathbf{#1}}a^{2} + 2 a + 2
\newcommand{\Bold}[1]{\mathbf{#1}}a^{2} + 2 a + 2
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0 2*a a + 1 a + 2 2 a 2*a + 2 2*a + 1 1 0 2*a a + 1 a + 2 2 a 2*a + 2 2*a + 1 1 |
We stellen de Zech log tabel op.
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\newcommand{\Bold}[1]{\mathbf{#1}}0
\newcommand{\Bold}[1]{\mathbf{#1}}0
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None
None |
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\newcommand{\Bold}[1]{\mathbf{#1}}a
\newcommand{\Bold}[1]{\mathbf{#1}}a
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\newcommand{\Bold}[1]{\mathbf{#1}}\left[2 a, a + 2, a, 2 a + 1\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[2 a, a + 2, a, 2 a + 1\right]
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Finite Field in u of size 2^3 Finite Field in u of size 2^3 |
(u^2 + 1)*x^2 + (u^2 + u + 1)*x + u^2 + u + 1 (u^2 + 1)*x^2 + (u^2 + u + 1)*x + u^2 + u + 1 |
0 [(u^2 + u, 1), (u^2 + u + 1, 1)] 0 [(u^2 + u, 1), (u^2 + u + 1, 1)] |
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\newcommand{\Bold}[1]{\mathbf{#1}}\left(u + 1\right) x^{3} + u x^{2} + u
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left(u + 1\right) x^{3} + u x^{2} + u
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\right] |
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\newcommand{\Bold}[1]{\mathbf{#1}}\left(u + 1\right) \cdot (x^{3} + \left(u^{2} + u + 1\right) x^{2} + u^{2} + u + 1)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(u + 1\right) \cdot (x^{3} + \left(u^{2} + u + 1\right) x^{2} + u^{2} + u + 1)
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\newcommand{\Bold}[1]{\mathbf{#1}}\left(1, x + u, \left(u^{2} + 1\right) x^{3} + \left(u + 1\right) x^{2} + u^{2}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(1, x + u, \left(u^{2} + 1\right) x^{3} + \left(u + 1\right) x^{2} + u^{2}\right)
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\newcommand{\Bold}[1]{\mathbf{#1}}23 \cdot 89
\newcommand{\Bold}[1]{\mathbf{#1}}23 \cdot 89
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\newcommand{\Bold}[1]{\mathbf{#1}}1936
\newcommand{\Bold}[1]{\mathbf{#1}}1936
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\newcommand{\Bold}[1]{\mathbf{#1}}x^{2047} + 1
\newcommand{\Bold}[1]{\mathbf{#1}}x^{2047} + 1
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