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     f ( x ) ≠constant ∈ K [ x ]

When f ( x )  has a root in K at least, K is algebraically closed field.

When K is algebraically closed field, arbitrary polynomial f ( x ) is separated to linear expression.

All the roots of f ( x ) 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 ( x ) | > | f ( 0 ) | is given under adequate longitude of r.

Closed disk  D = { x | |x| ≤ r } 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 ×…×K  is n-dimension affine space. Expression is Aⁿ(K) or Aⁿ_k.. Abbreviation is Aⁿ.

Finite polynomial     f₁, …, f_r ∈ K [ x₁, …, x_n ]

Common zero set of finite polynomial     V (  f₁, …, f_r  ) = { a ∈ Aⁿ | f₁ ( 0 ) = … = f_r ( a ) = 0 } ⊂ Aⁿ

The set is affine algebraic variety.

f₁, …, f_r  is defining equations.

20 <irreducible split>

Algebraic variety that is not empty V ⊂ Aⁿ is expressed by sum-set that is finite irreducible algebraic variety.

V = V₁  ⋃ … ⋃  V_r

The split is called irreducible split.

20* <Hilbert zero point theorem>

Algebraically closed field     K

n-variant polynomial ring     K [ x₁, …, x_n ]       

Ideal of K [ x₁, …, x_n ]         I

When 1 ∉ I is conditioned , V ( I ) ≠0 is realized

Tokyo September 24, 2007

Sekinan Research Field of Language