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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Algebraic field extension
In abstract algebra , a normal extension is an algebraic field extension L /K for which every irreducible polynomial over K that has a root in L splits into linear factors in L . This is one of the conditions for an algebraic extension to be a Galois extension . Bourbaki calls such an extension a quasi-Galois extension . For finite extensions , a normal extension is identical to a splitting field .
Definition [ edit ]
Let
L
/
K
{\displaystyle L/K}
be an algebraic extension (i.e., L is an algebraic extension of K ), such that
L
⊆
K
¯
{\displaystyle L\subseteq {\overline {K}}}
(i.e., L is contained in an algebraic closure of K ). Then the following conditions, any of which can be regarded as a definition of normal extension , are equivalent:
Every embedding of L in
K
¯
{\displaystyle {\overline {K}}}
over K induces an automorphism of L .
L is the splitting field of a family of polynomials in
K
[
X
]
{\displaystyle K[X ]}
.
Every irreducible polynomial of
K
[
X
]
{\displaystyle K[X ]}
that has a root in L splits into linear factors in L .
Other properties [ edit ]
Let L be an extension of a field K . Then:
If L is a normal extension of K and if E is an intermediate extension (that is, L ⊇ E ⊇ K ), then L is a normal extension of E .
If E and F are normal extensions of K contained in L , then the compositum EF and E ∩ F are also normal extensions of K .
Equivalent conditions for normality [ edit ]
Let
L
/
K
{\displaystyle L/K}
be algebraic. The field L is a normal extension if and only if any of the equivalent conditions below hold.
The minimal polynomial over K of every element in L splits in L ;
There is a set
S
⊆
K
[
x
]
{\displaystyle S\subseteq K[x ]}
of polynomials that each splits over L , such that if
K
⊆
F
⊊
L
{\displaystyle K\subseteq F\subsetneq L}
are fields, then S has a polynomial that does not split in F ;
All homomorphisms
L
→
K
¯
{\displaystyle L\to {\bar {K}}}
that fix all elements of K have the same image;
The group of automorphisms,
Aut
(
L
/
K
)
,
{\displaystyle {\text{Aut}}(L/K),}
of L that fix all elements of K , acts transitively on the set of homomorphisms
L
→
K
¯
{\displaystyle L\to {\bar {K}}}
that fix all elements of K .
Examples and counterexamples [ edit ]
For example,
Q
(
2
)
{\displaystyle \mathbb {Q} ({\sqrt {2}})}
is a normal extension of
Q
,
{\displaystyle \mathbb {Q} ,}
since it is a splitting field of
x
2
−
2.
{\displaystyle x^{2}-2.}
On the other hand,
Q
(
2
3
)
{\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})}
is not a normal extension of
Q
{\displaystyle \mathbb {Q} }
since the irreducible polynomial
x
3
−
2
{\displaystyle x^{3}-2}
has one root in it (namely,
2
3
{\displaystyle {\sqrt[{3}]{2}}}
), but not all of them (it does not have the non-real cubic roots of 2 ). Recall that the field
Q
¯
{\displaystyle {\overline {\mathbb {Q} }}}
of algebraic numbers is the algebraic closure of
Q
,
{\displaystyle \mathbb {Q} ,}
and thus it contains
Q
(
2
3
)
.
{\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}}).}
Let
ω
{\displaystyle \omega }
be a primitive cubic root of unity. Then since,
Q
(
2
3
)
=
{
a
+
b
2
3
+
c
4
3
∈
Q
¯
|
a
,
b
,
c
∈
Q
}
{\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})=\left.\left\{a+b{\sqrt[{3}]{2}}+c{\sqrt[{3}]{4}}\in {\overline {\mathbb {Q} }}\,\,\right|\,\,a,b,c\in \mathbb {Q} \right\}}
the map
{
σ
:
Q
(
2
3
)
⟶
Q
¯
a
+
b
2
3
+
c
4
3
⟼
a
+
b
ω
2
3
+
c
ω
2
4
3
{\displaystyle {\begin{cases}\sigma :\mathbb {Q} ({\sqrt[{3}]{2}})\longrightarrow {\overline {\mathbb {Q} }}\\a+b{\sqrt[{3}]{2}}+c{\sqrt[{3}]{4}}\longmapsto a+b\omega {\sqrt[{3}]{2}}+c\omega ^{2}{\sqrt[{3}]{4}}\end{cases}}}
is an embedding of
Q
(
2
3
)
{\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})}
in
Q
¯
{\displaystyle {\overline {\mathbb {Q} }}}
whose restriction to
Q
{\displaystyle \mathbb {Q} }
is the identity. However,
σ
{\displaystyle \sigma }
is not an automorphism of
Q
(
2
3
)
.
{\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}}).}
For any prime
p
,
{\displaystyle p,}
the extension
Q
(
2
p
,
ζ
p
)
{\displaystyle \mathbb {Q} ({\sqrt[{p}]{2}},\zeta _{p})}
is normal of degree
p
(
p
−
1
)
.
{\displaystyle p(p-1).}
It is a splitting field of
x
p
−
2.
{\displaystyle x^{p}-2.}
Here
ζ
p
{\displaystyle \zeta _{p}}
denotes any
p
{\displaystyle p}
th primitive root of unity . The field
Q
(
2
3
,
ζ
3
)
{\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}},\zeta _{3})}
is the normal closure (see below) of
Q
(
2
3
)
.
{\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}}).}
Normal closure [ edit ]
If K is a field and L is an algebraic extension of K , then there is some algebraic extension M of L such that M is a normal extension of K . Furthermore, up to isomorphism there is only one such extension that is minimal, that is, the only subfield of M that contains L and that is a normal extension of K is M itself. This extension is called the normal closure of the extension L of K .
If L is a finite extension of K , then its normal closure is also a finite extension.
See also [ edit ]
Citations [ edit ]
References [ edit ]
Lang, Serge (2002), Algebra , Graduate Texts in Mathematics , vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4 , MR 1878556
Jacobson, Nathan (1989), Basic Algebra II (2nd ed.), W. H. Freeman, ISBN 0-7167-1933-9 , MR 1009787
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Normal_extension&oldid=1221876268 "
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● T h i s p a g e w a s l a s t e d i t e d o n 2 M a y 2 0 2 4 , a t 1 4 : 3 4 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
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