Sekinan Table: April 2025

Premise of Algebraic Linguistics 3-1

　　 TANAKA Akio

1 <finitely generated>

Group G

Subset of G S

G is generated by finite set S. G is finitely generated.

2 <ascending chain rule>

Commutative ring A

Ideal of A

Ascending chain of I stops finitely.

The situation satisfies ascending chain rule.

3 <descending chain rule>

Commutative ring A

Ideal of A

Descending chain of I stops finitely.

The situation satisfies descending chain rule.

4 <maximum element>

Set defined by order X

Element of X a, x

x that is a < x does not exist.

a is maximum element.

5 <minimum element>

Set defined by order X

Element of X b, x

x that is b > x does not exist.

b is minimum element.

4 <Noetherian ring>

Commutative ring A that satisfies next equivalent conditions is Noetherian ring.

(1) A satisfies ascending chain rule on ideal.

(2) Ideal family of A has maximum element.

(3) Ideal of A is finitely generated.

5 <Artinian ring>

Commutative ring A that satisfies next equivalent conditions is Artininian ring.

(1) A satisfies descending chain rule on ideal.

(2) Ideal family of A has minimum element.

6 <module>

Additive group M

Ring A

M that has action of A is A module.

M satisfies next conditions.

(1) a ( x + y ) = ax + ay, ( a + b ) x = ax + bx

(2) a ( bx ) = ( ab ) x, 1x = x

7 <direct sum>

A module M, N

Structure of A module is given by set of product M ×N. The situation is expressed by M ⊕ N.

Direct sum of n-M, i.e. M ⊕M ⊕….⊕M is expressed by Mⁿ.

Tokyo September 23, 2007

Linguistic Premise Premise of Algebraic Linguistics 3

Premise of Algebraic Linguistics 3

　　 TANAKA Akio

Definition of <presheaf>

Topological space X

Arbitrary opened set U, V U < V

Commutative group F ( U )

Homomorphism τ_UV : F ( V ) → F ( U )

Given conditions

(1)

F ( 0 ) = { 0 }

(2)

τ_UV = id_U (Identity map)

(3)

U ⊂ V ⊂ W τ_UW =τ_UV oτ_VV

Presheaf of commutative ring on X { F ( U ), τ_UV}

Definition of <sheaf>

Presheaf F, G on X

Homomorphism ψ : F → G

Given conditions

(1)

Arbitrary opened set U

Homomorphism φ ( U ) : F ( U ) → G ( U )

(2)

Opened sets U < V

Below makes commutative diagram.

F ( V ) , G ( V ), F ( U ) , G ( U )

φ ( V ), φ.( U )

τ_UV,

(3)

F ( U ) ∋ s

Open covering U

{ U_i}_i_∈_I, r U_i, U = 0 i ∈ I ⇒ s = 0

(4)

Open covering U

{ U_i}_i_∈_I

F ( U ) ∋ s_ii ∈ I

rU_i ∩U_j U_i (s_i ) = r U_i∩U_j (s_j) (i, j ∈ I ) ⇒ rU_i, U (s) = s_i ( i ∈ I )

Presheaf is sheaf.

Tokyo September 17, 2007

Linguistic Premise Premise of Algebraic Linguistics 2-4

Premise of Algebraic Linguistics 2-4

　　 TANAKA Akio

32 <automorphism group>

Algebraic system X

Automorphism bijectional homomorphism from X to X

All the automorphism has productive composition. Automorphism group Expression is Aut ( X )

33 <operate>

Ring R

Automorphism group Aut ( X )

Homomorphism of group G → Aut( R )

G operates R.

Homomorphism is surjection. G separates R faithfully.

34 <normal separable extension>

Finite group G operates field L faithfully.

Invariant subfield of L k = L^G := { a ∈ L | σ( a ) = a ( ∀σ ∈ G )

Normal separable extension ( Galois extension ) [ L : k ] = | G |

35 <finite separable normal extension i.e. Galois extension>

Galois extension of L/k G = Gal ( L/k )

Subfield of G H

Invariant subfield of H L^H = { a ∈ L | σ ( a ) = a ∀ σ ∈ H }

Intermediate field of L/k E

H ( E )= { σ ∈ G | σ ( a ) = a ∀ a ∈ E }

36 <fundamental theorem of Galois theory>

Extension field is controlled by group theory.

Field of characteristic 0 k

Finite Galois extension K ⊃ k

Intermediate field of K ⊃ k M

Galois group Gal ( K/k )

Subgroup of Gal ( K/k ) H

Galois correspondence

(1) M →^φ Gal ( K/M )

(2) H →^ψ L^H

Extension M ⊃ k ⇔ Gal ( K/M ) is normal subfield of Gal ( K/k ).

Isomorphism Gal ( M/k ) ≅ Gal ( K/k ) / Gal ( K/M )

Tokyo September 22, 2007

Linguistic Premise Premise of Algebraic Linguistics 2-3

Premise of Algebraic Linguistics 2-3

　　 TANAKA Akio

25 <principal polynomial>

Commutative ring R

Elements of R a₀, a₁, …, a_n

Variant x

n-dimension polynomial over R (with R coefficient) a₀xⁿ + a₁xⁿ^-1 + … + a_ndegree( 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 = a^p+ b^p ( a + b ) ^q = a^q+ b^q

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

Linguistic Premise Premise of Algebraic Linguistics 2-2

Premise of Algebraic Linguistics 2-2

　　 TANAKA Akio

13 <extension field>

Field K, k K ⊃ k

Extension field K

Subfield k

Extension field is also expressed by K/k.

14 <extension dimension>

Extension field K/k

Extension dimension K dimension over k expressed by [ K : k ]

n-dimension extension [ K : k ] = n

15 <principal extension>

Field extension K/k

α ∈ K

Principal extension of k Minimum field containing k and α Expressed by k (α)

16 <transcendental and algebraic>

Field extension K/k

Polynomial ring k [ X ]

Homomorphism φ= φ_α : k [ X ] → K, φ( f ( X ) ) = f (α)

Ker (φ) = ( 0 ) α is transcendental over k.

Ker (φ) ≠ ( 0 ) α is algebraic over k.

17 <irreducible polynomial and principal polynomial>

Ker (φ) = ( f ( X ) ) uniquely determines principal polynomial that is expressed by Irr_k ( α ).

18 <algebraic extension>

Arbitrary element of K is algebraic over k, K/k is algebraic extension.

19 <chain rule of extension dimension>

Finite extension k ⊂ K ⊂ L

[ L : k ] = [ L : K ] [ K : k ]

20 <algebraically closed field and algebraic closure>

Field Ω

Arbitrary not constant f ∈ Ω f ( α ) = 0 and α⊂Ω

Ω is algebraic closure.

21 <root of f ( X )>

Extension field K/k

Set of K’s k-isomorphism Aut_k( K )

K = k ( α)

α’s irreducible polynomial f ( X )

σ ∈ Aut_k( K ) is determined by σ ( α ) that is root of f ( X ).

| Aut_k( K ) | = # { a ∈ K | f ( a ) = 0 }

22 <normal extension>

Finite extension field K/k

All the roots of K’s irreducible polynomial against arbitrary element α has roots of K. K/k is normal extension.

23 <splitting field>

Polynomial f ( X ) ∈ k [ X ]

All the roots of f ( X ) adjoining to k f splitting field over k

23*

K/k is normal extension. ⇔ K is f ( X ) ∈ k [ X ]’s splitting field over k.

24 <Galois group>

Splitting field of f ∈ k [ X ]

Aut_k( K ) is f’s Galois group. Expression is Gal ( f ).

Tokyo September 21, 2007