13 <extension field>
Field K, k K ⊃ k
Extension field K
Subfield k
Extension field is also expressed by K/k.
14 <extension dimension>
Extension field K/k
Extension dimension K dimension over k expressed by [ K : k ]
n-dimension extension [ K : k ] = n
15 <principal extension>
Field extension K/k
α ∈ K
Principal extension of k Minimum field containing k and α Expressed by k (α)
16 <transcendental and algebraic>
Field extension K/k
Polynomial ring k [ X ]
Homomorphism φ= φα : k [ X ] → K, φ( f ( X ) ) = f (α)
Ker (φ) = ( 0 ) α is transcendental over k.
Ker (φ) ≠ ( 0 ) α is algebraic over k.
17 <irreducible polynomial and principal polynomial>
Ker (φ) = ( f ( X ) ) uniquely determines principal polynomial that is expressed by Irrk ( α ).
18 <algebraic extension>
Arbitrary element of K is algebraic over k, K/k is algebraic extension.
19 <chain rule of extension dimension>
Finite extension k ⊂ K ⊂ L
[ L : k ] = [ L : K ] [ K : k ]
20 <algebraically closed field and algebraic closure>
Field Ω
Arbitrary not constant f ∈ Ω f ( α ) = 0 and α⊂Ω
Ω is algebraic closure.
21 <root of f ( X )>
Extension field K/k
Set of K’s k-isomorphism Autk ( K )
K = k ( α)
α’s irreducible polynomial f ( X )
σ ∈ Autk ( K ) is determined by σ ( α ) that is root of f ( X ).
| Autk ( K ) | = # { a ∈ K | f ( a ) = 0 }
22 <normal extension>
Finite extension field K/k
All the roots of K’s irreducible polynomial against arbitrary element α has roots of K. K/k is normal extension.
23 <splitting field>
Polynomial f ( X ) ∈ k [ X ]
All the roots of f ( X ) adjoining to k f splitting field over k
23*
K/k is normal extension. ⇔ K is f ( X ) ∈ k [ X ]’s splitting field over k.
24 <Galois group>
Splitting field of f ∈ k [ X ]
Autk ( K ) is f’s Galois group. Expression is Gal ( f ).
Tokyo September 21, 2007
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