Monday 5 January 2015

Linguistic Premise Premise of Algebraic Linguistics 3-4 / September 24, 2007

Linguistic Premise

 Premise of Algebraic Linguistics 3-4

    TANAKA Akio

17 <algebraically closed field>
Field     K
Polynomial over K     x ) constant  K [ ]
When f x )  has a root in K at least, K is algebraically closed field.
When K is algebraically closed field, arbitrary polynomial f ( ) is separated to linear expression.
All the roots of f ( ) are elements of K.

18 <fundamental theorem of algebra>
Complex field C is algebraically closed field.
[Proof outline]
x over circumference that has radius r
| f ( ) | > | f ( 0 ) | is given under adequate longitude of r.
Closed disk  D = { x | |x } is compact.
α  D that minimizes continuous function | f ( x ) | exists.
α is in D.
f ( α) = 0
If f ( α 0, | f ( α) | is not the minimum value of f ( x ).

19 <affine space>
Field    K
Direct product set  Kn = K ×× is n-dimension affine space. Expression is An(K) or Ank.. Abbreviation is An.
Finite polynomial     f1, …, fr  K [ x1, …, xn ]
Common zero set of finite polynomial     (  f1, …, f ) = {  An | f1 ( 0 ) = … = fr a ) = 0 }  An
The set is affine algebraic variety.
f1, …, f is defining equations.

20 <irreducible split>
Algebraic variety that is not empty  An is expressed by sum-set that is finite irreducible algebraic variety.
V ⋃ …   Vr
The split is called irreducible split.

20* <Hilbert zero point theorem>
Algebraically closed field     K
n-variant polynomial ring     x1, …, xn ]       
Ideal of x1, …, xn ]         I
When 1  is conditioned , I ) 0 is realized

Tokyo September 24, 2007
Sekinan Research Field of Language

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