25 <principal polynomial>
Commutative ring R
Elements of R a0, a1, …, an
Variant x
n-dimension polynomial over R (with R 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 r < deg g
26 <minimal polynomial>
Extension field K/k
K’ element αis algebraic over k . polynomial f ( x ) ≠ 0 ∈ k [ x ] f ( 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 k is separable. K/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. K = k (α)
α minimal polynomial over k f ( X )
Intermediate field of K/k L
K = L (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 g ( X ) ‘s coefficient to k L’ ⊂ L g ( X ) 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 ( X )
The coefficient added to k makes L.
f ( X ) is finite.
Number of intermediate field is finite.
29 <Frobenius map>
Ring A
p ∈ A
p = 0
Map F : A → A F ( a ) = 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.
( 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/k is normal and separable.
Tokyo September 22, 2007
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