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 Ank.. Abbreviation is An.
Finite polynomial f1, …, fr ∈ K [ x1, …, xn ]
Common zero set of finite polynomial V ( f1, …, fr ) = { a ∈ An | f1 ( 0 ) = … = fr ( a ) = 0 } ⊂ An
The set is affine algebraic variety.
f1, …, fr 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 ⋃ … ⋃ Vr
The split is called irreducible split.
20* <Hilbert zero point theorem>
Algebraically closed field K
n-variant polynomial ring K [ x1, …, xn ]
Ideal of K [ x1, …, xn ] I
When 1 ∉ I is conditioned , V ( I ) ≠0 is realized
Tokyo September 24, 2007
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