Algebraic Fields and Computable Categoricity Russell Miller & Alexandra Shlapentokh Queens College & CUNY Graduate Center East Carolina University New York, NY Greenville, NC. George Washington University Logic Seminar 19 November 2010 (Some work joint with Denis Hirschfeldt, University of Chicago, and Ken Kramer, CUNY.) Slides available at qc.edu/˜rmiller/slides.html Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 1 / 19
Computable Categoricity Defn. A computable structure A is computably categorical if for each computable B ∼ = A there is a computable isomorphism from A onto B . Examples : (Dzgoev, Goncharov; Remmel; Lempp, McCoy, M., Solomon) A linear order is computably categorical iff it has only finitely many adjacencies. A Boolean algebra is computably categorical iff it has only finitely many atoms. An ordered Abelian group is computably categorical iff it has finite rank ( ≡ basis as Z -module). For trees, the known criterion is recursive in the height and not easily stated! Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 2 / 19
Computably Categorical Fields The following fields are all computably categorical: Q . All finitely generated extensions of Q or F p . Every algebraically closed field of finite transcendence degree over Q or F p . All normal algebraic extensions of Q or F p . Some (but not all) non-normal algebraic extensions of Q or F p . Certain fields (but not very many!) of infinite transcendence degree over Q . (Miller-Schoutens.) Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 3 / 19
Relative Computable Categoricity Defn. A computable structure A is relatively computably categorical if for each B ∼ = A with domain ω , there is an isomorphism from A onto B which is computable from an oracle for B . Clearly this implies computable categoricity – but the converse is false! Certain computably categorical structures are not relatively computably categorical. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 4 / 19
Scott Families Defn. A Scott family for a structure A is a set Σ of formulas ψ ( x 0 , . . . , x n ,� c ) , over a fixed finite tuple � c of parameters from A , such that a ∈ A <ω , some ψ ∈ Σ has | For all � = A ψ ( � a ,� c ) . b ∈ A n satisfy the same ψ ∈ Σ , then some α ∈ Aut ( A ) has a ,� If � α ( a i ) = b i for all i ≤ n . s s s s s s s s s s s s s s s s c Example: . . . s s s s s Thm. (Ash-Knight-Manasse-Slaman; Chisholm) A computable structure A is relatively computably categorical iff A has a computably enumerable Scott family of existential formulas. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 5 / 19
Algebraic Fields with Splitting Algorithms Definitions A field is algebraic if it is an algebraic extension of its prime subfield (either Q or F p ). A computable field F has a splitting algorithm if its splitting set S F (or equivalently its root set R F ) is computable: S F = { p ∈ F [ X ] : p factors properly in F [ X ] } R F = { p ∈ F [ X ] : ( ∃ a ∈ F ) p ( a ) = 0 } Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 6 / 19
Algebraic Fields with Splitting Algorithms Definitions A field is algebraic if it is an algebraic extension of its prime subfield (either Q or F p ). A computable field F has a splitting algorithm if its splitting set S F (or equivalently its root set R F ) is computable: S F = { p ∈ F [ X ] : p factors properly in F [ X ] } R F = { p ∈ F [ X ] : ( ∃ a ∈ F ) p ( a ) = 0 } Facts: All finite algebraic extensions of Q and F p have splitting algorithms, uniformly in their generators. An algebraic field F has a splitting algorithm iff all computable fields isomorphic to F have splitting algorithms. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 6 / 19
Orbit Relations for Fields Definition For a computable field F , the full orbit relation A F for F is the set: {� a 1 , . . . , a n ; b 1 , . . . , b n � : ( ∃ σ ∈ Aut ( F ))( ∀ i ) σ ( a i ) = b i } ⊆ ∪ n F 2 n . For algebraic F , by the Effective Theorem of the Primitive Element, A F is computably isomorphic to the orbit relation B F of F , defined by the action of Aut ( F ) : B F = {� a ; b � ∈ F 2 : ( ∃ σ ∈ Aut ( F )) σ ( a ) = b } . Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 7 / 19
Orbit Relations for Fields Definition For a computable field F , the full orbit relation A F for F is the set: {� a 1 , . . . , a n ; b 1 , . . . , b n � : ( ∃ σ ∈ Aut ( F ))( ∀ i ) σ ( a i ) = b i } ⊆ ∪ n F 2 n . For algebraic F , by the Effective Theorem of the Primitive Element, A F is computably isomorphic to the orbit relation B F of F , defined by the action of Aut ( F ) : B F = {� a ; b � ∈ F 2 : ( ∃ σ ∈ Aut ( F )) σ ( a ) = b } . For algebraic F ⊇ Q in general, B F is Π 0 2 : � a ; b � ∈ B F iff ( ∀ q ∈ Q [ X , Y ]) [ q ( a , Y ) ∈ R F ⇐ ⇒ q ( b , Y ) ∈ R F ] . However, when F has a splitting algorithm, B F becomes Π 0 1 . (And when F ⊇ Q is a normal algebraic extension, B F is computable.) Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 7 / 19
Computable Categoricity Theorem (MS 2010) Let F be a computable algebraic field with a splitting algorithm. Then F is computably categorical iff B F is computable. Since F has a splitting algorithm, B F is Π 0 1 , so the complexity of this condition is Σ 0 3 in indices for F and its splitting algorithm. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 8 / 19
Computable Categoricity Theorem (MS 2010) Let F be a computable algebraic field with a splitting algorithm. Then F is computably categorical iff B F is computable. Since F has a splitting algorithm, B F is Π 0 1 , so the complexity of this condition is Σ 0 3 in indices for F and its splitting algorithm. Corollary A computable algebraic field with a splitting algorithm is computably categorical iff it is relatively computably categorical. (The proof below relativizes easily.) Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 8 / 19
B F Computable = ⇒ F Computably Categorical Sketch of Proof: Suppose we have defined f s : F s = Q ( x 0 , . . . , x s ) → E , where F = { x 0 , x 1 , . . . } , and F s ⊆ F s + 1 are both normal within F . Assume f s extends to an isomorphism ψ : F → E . Find a primitive generator a ∈ F of F s + 1 , and find its minimal polynomial p ( X ) ∈ F s [ X ] . Let a = y 1 , y 2 , . . . , y d be all its roots in F . Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 9 / 19
B F Computable = ⇒ F Computably Categorical Sketch of Proof: Suppose we have defined f s : F s = Q ( x 0 , . . . , x s ) → E , where F = { x 0 , x 1 , . . . } , and F s ⊆ F s + 1 are both normal within F . Assume f s extends to an isomorphism ψ : F → E . Find a primitive generator a ∈ F of F s + 1 , and find its minimal polynomial p ( X ) ∈ F s [ X ] . Let a = y 1 , y 2 , . . . , y d be all its roots in F . For each j ≤ d with � a , y j � / ∈ B F , find some q j ∈ F s [ X , Y ] with q j ( a , Y ) ∈ R F & q j ( y j , Y ) / ∈ R F . Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 9 / 19
B F Computable = ⇒ F Computably Categorical Sketch of Proof: Suppose we have defined f s : F s = Q ( x 0 , . . . , x s ) → E , where F = { x 0 , x 1 , . . . } , and F s ⊆ F s + 1 are both normal within F . Assume f s extends to an isomorphism ψ : F → E . Find a primitive generator a ∈ F of F s + 1 , and find its minimal polynomial p ( X ) ∈ F s [ X ] . Let a = y 1 , y 2 , . . . , y d be all its roots in F . For each j ≤ d with � a , y j � / ∈ B F , find some q j ∈ F s [ X , Y ] with q j ( a , Y ) ∈ R F & q j ( y j , Y ) / ∈ R F . Then find all roots z 1 , . . . , z d ∈ E of the image p ( X ) of p ( X ) under f s . Define f s + 1 ( a ) to be any z k for which all the polynomials q j ( z k , Y ) have roots in E . Then f s ⊆ f s + 1 and � a , ψ − 1 ( z k ) � ∈ B F , so f s + 1 must extend to the isomorphism ψ ◦ σ : F → E , where σ ∈ Aut ( F ) has σ ( a ) = ψ − 1 ( z k ) and ( ∀ i ) σ ( x i ) = x i . By iterating, we get a computable isomorphism. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 9 / 19
F Computably Categorical = ⇒ B F Computable Proof: Here we assume that F is computably categorical, and build a computable E ∼ = F . In doing so, whenever possible, we build E so that ϕ e will not be an isomorphism. (This uses a priority construction, based on the values e .) For the least e such that ϕ e defies all our attempts, the isomorphism ϕ e will allow us to compute B F . Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 10 / 19
Recommend
More recommend
