Monday 5 January 2015

Linguistic Premise Premise of Algebraic Linguistics 2-3 / September 22, 2007

Linguistic Premise

 Premise of Algebraic Linguistics 2-3

    TANAKA Akio

25 <principal polynomial>
Commutative ring     R
Elements of R     a0a1, …, an
Variant    x
n-dimension polynomial over (with coefficient)     a0xn + a1xn-1 + … + an     degree( deg ) = n
Principal polynomial     Polynomial with maximum coefficient is 1.
Polynomial     f
Principal polynomial     g
f = qg + r    deg < deg g

26 <minimal polynomial>
Extension field     K/k
K’ element αis algebraic over k .    polynomial   0   k [ x ]     α ) = 0
What k-coefficient irreducible polynomial that has root α is minimal polynomial.

27 <separable extension>
Extension field     K/k
Arbitrary     α  K
Root of minimal polynomial over is separable.      K/is separable extension.

28 <intermediate field>
Finite extension field     K/k
What K/k is principal expansion is equivalent to what K/k ‘s intermediate field is finite.
[Proof outline]
K/k is principal extension.    (α)
α minimal polynomial over     X )
Intermediate field of K/k      L
K = (α)
α irreducible polynomial over L     g ( X )  L ( X )    L ( X )  is divided by f ( X ).
Expansion dimension [ K : L ] = deg g ( X )
Field that adds all the X ) ‘s coefficient to k     L’  L   g ) is irreducible at L’ ( X ).
[ K : L’ ] = deg g ( X ) = [ K : L ]    L = L
Arbitrary L
Expansion field     f ( X )
Factor    g ( X )
Coefficient of g ( )
The coefficient added to makes L.
f ( ) is finite.
Number of intermediate field is finite.

29 <Frobenius map>
Ring     A
p  A
p = 0
Map     F A      F ( ) = a p
F is Frobenius map of ring A.

29*
Frobenius map is homomorphism of ring.
[Proof outline]
From binomial theorem binomial coefficient is 0 in ring A.
b ) p = ap bp   ( a b ) q = aq bq

30 <perfect field>
Perfect field has only separable fields.

31 <Galois extension>
Finite extension field     L/k
Galois extension      L/is normal and separable.

Tokyo September 22, 2007
Sekinan Research Field of Language

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