Monday 5 January 2015

Linguistic Premise Premise of Algebraic Linguistics 2-2 / September 21, 2007


Linguistic Premise

 Premise of Algebraic Linguistics 2-2

    TANAKA Akio

13 <extension field>
Field     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     φφα : k [ X ]  K φf ( 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 is algebraic over kK/k is algebraic extension.

19 <chain rule of extension dimension>
Finite extension      K  L
] = [ L : ] [ K : k ]

20 <algebraically closed field and algebraic closure>
Field     Ω
Arbitrary not constant f  Ω   f ( α ) = 0 and αΩ
Ω is algebraic closure.

21 <root of f ( X )>
Extension field     K/k
Set of K’s k-isomorphism     Aut)
k ( α)
α’s irreducible polynomial     f ( X )
σ ∈ Aut) is determined by σ ( α ) that is root of f ( ).
| Aut) | = # {  K  | f ( ) = 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 ) adjoining to      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  ]
Aut( ) is f’s Galois group. Expression is Gal ( f ).

Tokyo September 21, 2007
Sekinan Research Field of Language

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