De lineaire afbeelding
L
:
V
→
W
{\displaystyle L:V\to W}
v
1
,
…
,
v
n
{\displaystyle v_{1},\ldots ,v_{n}}
V
{\displaystyle V}
w
1
,
…
,
w
m
{\displaystyle w_{1},\ldots ,w_{m}}
W
{\displaystyle W}
m
×
n
{\displaystyle m\times n}
A
{\displaystyle A}
L
v
1
,
…
,
L
v
n
{\displaystyle Lv_{1},\ldots ,Lv_{n}}
n
{\displaystyle n}
A
{\displaystyle A}
L
v
k
=
∑
r
=
1
m
c
k
r
w
r
{\displaystyle Lv_{k}=\sum _{r=1}^{m}c_{kr}w_{r}}
A
r
k
=
c
k
r
{\displaystyle A_{rk}=c_{kr}}
A
=
(
(
c
11
,
…
,
c
1
m
)
,
…
,
(
c
n
1
,
…
,
c
n
m
)
)
=
c
T
{\displaystyle A=((c_{11},\ldots ,c_{1m}),\ldots ,(c_{n1},\ldots ,c_{nm}))=c^{T}}
Een
1
×
n
{\displaystyle 1\times n}
A
{\displaystyle A}
A
=
(
(
c
11
)
,
…
,
(
c
n
1
)
)
=
c
T
{\displaystyle A=((c_{11}),\ldots ,(c_{n1}))=c^{T}}
En een
m
×
1
{\displaystyle m\times 1}
A
{\displaystyle A}
A
=
(
(
c
11
,
…
,
c
1
m
)
)
=
c
T
{\displaystyle A=((c_{11},\ldots ,c_{1m}))=c^{T}}
Het beeld
L
x
=
L
(
x
1
v
1
+
…
+
x
n
v
n
)
=
∑
i
=
1
m
x
i
L
(
v
i
)
=
{\displaystyle Lx=L(x_{1}v_{1}+\ldots +x_{n}v_{n})=\sum _{i=1}^{m}x_{i}L(v_{i})=}
=
∑
i
=
1
m
x
i
∑
r
=
1
m
c
i
r
w
r
=
∑
r
=
1
m
ξ
r
w
r
{\displaystyle =\sum _{i=1}^{m}x_{i}\sum _{r=1}^{m}c_{ir}w_{r}=\sum _{r=1}^{m}\xi _{r}w_{r}}
dus
ξ
r
=
∑
i
=
1
m
x
i
c
i
r
=
∑
i
=
1
m
A
r
i
x
i
{\displaystyle \xi _{r}=\sum _{i=1}^{m}x_{i}c_{ir}=\sum _{i=1}^{m}A_{ri}x_{i}}
L
=
G
F
(
25
)
,
K
=
G
F
(
5
)
{\displaystyle L=\mathrm {GF} (25),\,K=\mathrm {GF} (5)}
Een endomorfisme
φ
{\displaystyle \varphi }
L
{\displaystyle L}
φ
(
x
+
y
)
=
φ
(
x
)
+
φ
(
y
)
{\displaystyle \varphi (x+y)=\varphi (x)+\varphi (y)}
dus
φ
(
m
x
)
=
m
φ
(
x
)
{\displaystyle \varphi (mx)=m\varphi (x)}
en
φ
(
x
)
=
φ
(
a
+
b
2
)
=
a
φ
(
1
)
+
b
φ
(
2
)
=
a
α
k
+
b
α
m
{\displaystyle \varphi (x)=\varphi (a+b{\sqrt {2}})=a\varphi (1)+b\varphi ({\sqrt {2}})=a\alpha ^{k}+b\alpha ^{m}}
Endomorfisme wordt bepaald door:
φ
(
1
)
=
α
k
{\displaystyle \varphi (1)=\alpha ^{k}}
en
φ
(
2
)
=
α
m
{\displaystyle \varphi ({\sqrt {2}})=\alpha ^{m}}
Er zijn dus 625 endomorfismen
φ
(
0
)
=
0
{\displaystyle \varphi (0)=0}
φ
(
m
)
=
m
φ
(
1
)
=
m
α
k
{\displaystyle \varphi (m)=m\varphi (1)=m\alpha ^{k}}
dus
φ
(
α
)
=
φ
(
2
+
2
)
=
φ
(
2
)
+
φ
(
2
)
=
2
α
k
+
α
m
{\displaystyle \varphi (\alpha )=\varphi (2+{\sqrt {2}})=\varphi (2)+\varphi ({\sqrt {2}})=2\alpha ^{k}+\alpha ^{m}}
Frobenius:
φ
(
x
)
=
x
5
{\displaystyle \varphi (x)=x^{5}}
dus
φ
(
1
)
=
1
{\displaystyle \varphi (1)=1}
en
φ
(
2
)
=
4
2
=
α
21
{\displaystyle \varphi ({\sqrt {2}})=4{\sqrt {2}}=\alpha ^{21}}
NB
y
24
=
1
{\displaystyle y^{24}=1}
x
24
m
+
d
=
(
x
m
)
24
x
d
=
x
d
;
0
≤
d
<
24
{\displaystyle x^{24m+d}=(x^{m})^{24}x^{d}=x^{d};\,0\leq d<24}
x
d
=
x
5
m
+
v
=
x
5
m
x
v
=
(
x
m
)
5
x
v
=
(
x
m
)
∗
x
v
;
0
≤
m
,
v
<
5
{\displaystyle x^{d}=x^{5m+v}=x^{5m}x^{v}=(x^{m})^{5}x^{v}=(x^{m})^{*}x^{v};\,0\leq m,v<5}
bv
x
59
=
x
24
⋅
2
+
11
=
(
x
2
)
24
x
11
=
x
11
;
0
≤
d
<
24
{\displaystyle x^{59}=x^{24\cdot 2+11}=(x^{2})^{24}x^{11}=x^{11};\,0\leq d<24}
x
11
=
x
5
⋅
2
+
1
=
x
5
⋅
2
x
1
=
(
x
2
)
5
x
1
=
(
x
2
)
∗
x
;
0
≤
m
,
v
<
5
{\displaystyle x^{11}=x^{5\cdot 2+1}=x^{5\cdot 2}x^{1}=(x^{2})^{5}x^{1}=(x^{2})^{*}x;\,0\leq m,v<5}
(
a
+
b
2
)
5
m
=
(
a
+
(
−
1
)
m
b
2
)
{\displaystyle (a+b{\sqrt {2}})^{5m}=(a+(-1)^{m}b{\sqrt {2}})}
(
a
−
b
2
)
=
(
a
+
b
2
)
∗
{\displaystyle (a-b{\sqrt {2}})=(a+b{\sqrt {2}})^{*}}
