In body theory , the norm of a body expansion is a special mapping assigned to the expansion. It maps every element of the larger body onto the smaller body.
This norm concept differs significantly from the concept of the norm of a normalized vector space , so it is sometimes called a body norm in contrast to the vector norm .
definition
Let it be a finite expansion of the body . A fixed element defines a - linear mapping
L.
/
K
{\ displaystyle L / K}
a
∈
L.
{\ displaystyle a \ in L}
K
{\ displaystyle K}
L.
→
L.
,
x
↦
a
x
.
{\ displaystyle L \ to L, \ quad x \ mapsto ax.}
Its determinant is called the norm of , written . She is an element of ; so the norm is a map
a
{\ displaystyle a}
N
L.
/
K
(
a
)
{\ displaystyle N_ {L / K} (a)}
K
{\ displaystyle K}
N
L.
/
K
:
L.
→
K
,
a
↦
N
L.
/
K
(
a
)
.
{\ displaystyle N_ {L / K} \ colon L \ to K, \ quad a \ mapsto N_ {L / K} (a).}
properties
Exactly applies to .
a
=
0
{\ displaystyle a = 0}
N
L.
/
K
(
a
)
=
0
{\ displaystyle N_ {L / K} (a) = 0}
The norm is multiplicative, i.e. H.
N
L.
/
K
(
a
b
)
=
N
L.
/
K
(
a
)
⋅
N
L.
/
K
(
b
)
{\ displaystyle N_ {L / K} (down) = N_ {L / K} (a) \ cdot N_ {L / K} (b)}
for everyone .
a
,
b
∈
L.
{\ displaystyle a, b \ in L}
Restricted to the multiplicative groups , the norm is therefore a homomorphism
N
L.
/
K
:
L.
×
→
K
×
.
{\ displaystyle N_ {L / K} \ colon L ^ {\ times} \ to K ^ {\ times}.}
If there is a further finite expansion of the body, then one has the three norm functions and , which are related in the following, called the transitivity of the norm :
M.
/
L.
{\ displaystyle M / L}
N
L.
/
K
,
N
M.
/
L.
{\ displaystyle N_ {L / K}, N_ {M / L}}
N
M.
/
K
{\ displaystyle N_ {M / K}}
N
M.
/
K
(
a
)
=
N
L.
/
K
(
N
M.
/
L.
(
a
)
)
{\ displaystyle N_ {M / K} (a) = N_ {L / K} (N_ {M / L} (a))}
for everyone .
a
∈
M.
{\ displaystyle a \ in M}
Is , then applies .
a
∈
K
{\ displaystyle a \ in K}
N
L.
/
K
(
a
)
=
a
[
L.
:
K
]
{\ displaystyle N_ {L / K} (a) = a ^ {[L: K]}}
If the minimal polynomial is of degree , the absolute term of and , then:
a
∈
L.
{\ displaystyle a \ in L}
f
∈
K
[
X
]
{\ displaystyle f \ in K [X]}
d
{\ displaystyle d}
a
0
∈
K
{\ displaystyle a_ {0} \ in K}
f
{\ displaystyle f}
r
=
[
L.
:
K
(
a
)
]
{\ displaystyle r = [L: K (a)]}
N
L.
/
K
(
a
)
=
(
-
1
)
d
r
a
0
r
{\ displaystyle N_ {L / K} (a) = (- 1) ^ {dr} a_ {0} ^ {r}}
Is a finite field extension with , with the number of elements in the set of all -Homomorphismen of the algebraic closure of was. Then applies to every element
L.
/
K
{\ displaystyle L / K}
[
L.
:
K
]
=
q
r
{\ displaystyle [L: K] = qr}
r
{\ displaystyle r}
σ
{\ displaystyle \ sigma}
Hom
K
(
L.
,
K
¯
)
{\ displaystyle \ operatorname {Hom} _ {K} (L, {\ bar {K}})}
K
{\ displaystyle K}
L.
{\ displaystyle L}
K
¯
{\ displaystyle {\ bar {K}}}
K
{\ displaystyle K}
a
∈
L.
{\ displaystyle a \ in L}
N
L.
/
K
(
a
)
=
(
∏
i
=
1
r
σ
i
(
a
)
)
q
{\ displaystyle N_ {L / K} (a) = \ left (\, \ prod _ {i = 1} ^ {r} \ sigma _ {i} (a) \ right) ^ {q}}
If in particular is Galois with Galois group , this means
L.
/
K
{\ displaystyle L / K}
Gal
(
L.
/
K
)
{\ displaystyle \ operatorname {Gal} (L / K)}
N
L.
/
K
(
a
)
=
∏
σ
∈
Gal
(
L.
/
K
)
σ
(
a
)
.
{\ displaystyle N_ {L / K} (a) = \ prod _ {\ sigma \ in \ operatorname {Gal} (L / K)} \ sigma (a).}
Examples
N
C.
/
R.
(
a
+
i
b
)
=
σ
1
(
a
+
i
b
)
σ
2
(
a
+
i
b
)
=
i
d
(
a
+
i
b
)
(
a
+
i
b
)
¯
=
(
a
+
i
b
)
(
a
-
i
b
)
=
a
2
+
b
2
{\ displaystyle N _ {\ mathbb {C} / \ mathbb {R}} (a + ib) = \ sigma _ {1} (a + ib) \ sigma _ {2} (a + ib) = id (a + ib) {\ overline {(a + ib)}} = (a + ib) (a-ib) = a ^ {2} + b ^ {2}}
.
The norm of is the picture
Q
(
2
)
/
Q
{\ displaystyle \ mathbb {Q} ({\ sqrt {2}}) / \ mathbb {Q}}
a
+
b
2
↦
a
2
-
2
b
2
{\ displaystyle a + b {\ sqrt {2}} \ mapsto a ^ {2} -2b ^ {2}}
for .
a
,
b
∈
Q
{\ displaystyle a, b \ in \ mathbb {Q}}
The norm of is the picture
F.
q
n
/
F.
q
{\ displaystyle \ mathbb {F} _ {q ^ {n}} / \ mathbb {F} _ {q}}
x
↦
x
1
+
q
+
q
2
+
...
+
q
n
-
1
{\ displaystyle x \ mapsto x ^ {1 + q + q ^ {2} + \ ldots + q ^ {n-1}}}
.
See also
Individual evidence
^ Bosch, Algebra 5th edition, 2004, p. 196ff
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">