Algebraic Fields and Computable Categoricity

Transcription

1 Algebraic Fields and Computable Categoricity Russell Miller & Alexandra Shlapentokh Queens College & CUNY Graduate Center New York, NY East Carolina University 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

2 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

3 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

4 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

5 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 For all a A <ω, some ψ Σ has = A ψ( a, c). If a, b A n satisfy the same ψ Σ, then some α Aut(A) has α(a i ) = b i for all i n. Example: c... 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

6 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

7 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: Facts: S F = {p F[X] : p factors properly in F[X]} R F = {p F[X] : ( a F) p(a) = 0} 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

8 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 2n. 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

9 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 2n. 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

10 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

11 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

12 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

13 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

14 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

15 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

16 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. At each stage s + 1, we look for the least e such that for some a, b F s, ϕ e,s (a) and a, b B Fs, yet a, b / B Fs+1. (Essentially we search for q Q[X, Y ] such that q(b, Y ) has a root in F and q(a, Y ) does not.) Then, when building the extension E s+1 of E s, we add a root of q(ϕ e (a), Y ), so that our isomorphism F s+1 E s+1 has b ϕ e (a), and no isomorphism F E has a ϕ e (a). Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 10 / 19

17 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. At each stage s + 1, we look for the least e such that for some a, b F s, ϕ e,s (a) and a, b B Fs, yet a, b / B Fs+1. (Essentially we search for q Q[X, Y ] such that q(b, Y ) has a root in F and q(a, Y ) does not.) Then, when building the extension E s+1 of E s, we add a root of q(ϕ e (a), Y ), so that our isomorphism F s+1 E s+1 has b ϕ e (a), and no isomorphism F E has a ϕ e (a). If ϕ e : F E is an isomorphism, with e minimal, then ( s s 0 ) [ϕ e,s (a) = ( b)[ a, b B Fs = a, b B F ]]. So B F is c.e., as well as Π 0 1. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 10 / 19

18 Algebraic Fields Without Splitting Algorithms Theorem (easy corollary of Ash-Knight-Manasse-Slaman) Let F be a computable, relatively computably categorical, algebraic field. Then the orbit relation B F is computably enumerable. This generalizes our theorem on fields with splitting algorithms, since for those fields, B F is automatically Π 0 1. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 11 / 19

19 Algebraic Fields Without Splitting Algorithms Theorem (easy corollary of Ash-Knight-Manasse-Slaman) Let F be a computable, relatively computably categorical, algebraic field. Then the orbit relation B F is computably enumerable. This generalizes our theorem on fields with splitting algorithms, since for those fields, B F is automatically Π 0 1. However, if F has no splitting algorithm, then B F can be c.e., or even computable, with F not computably categorical. Example: Begin to build E = F = Q(θ 0 ) with θ 3 0 = 2. If ϕ e(θ 0 ) = θ 0, then adjoin to E and F two more cube roots θ 1, θ 2 of 2. Also adjoin to E a square root of θ 0, and to F a square root of θ 1. Then ϕ e : E F is not an isomorphism, yet B E and B F remain computable. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 11 / 19

20 Full Construction: B F T, but F not C.C. Lemma For every Galois extension Q E and every d > 1, there exists a monic f (X) Z[X] of degree d such that Gal(K /Q) = S d and the splitting field K of f (X) over Q is linearly disjoint from E. Corollary There is a computable sequence f 0, f 1,... in Z[X] whose splitting fields K i each have Galois group S 7 over Q and such that each K i is linearly disjoint from the compositum of all K j (j i). Use this sequence to build F and F. For distinct roots r 1, r 2, r 3, r 4 of K i, first adjoin (r 1 + r 2 ) to Q in both F and F. If ϕ i (r 1 + r 2 ) = (r 1 + r 2 ), then adjoin (r 3 + r 4 ) to both fields, r 1 to F, and r 3 to F. Then F = F, but not via ϕ i. However, F is rigid (except for interchanging r 1 with r 2, if they entered F). Thus B F is computable. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 12 / 19

21 The Isomorphism Tree Let F = F be computable algebraic fields, and let z 1, z 2, z 3,... be a sequence of elements generating F. For simplicity, assume Q = Q(z 0 ) Q(z 1 ) Q(z 2 ), with z 0 = 1. Compute polynomials f i+1 Q[Y, Z ] s.t. f i+1 (z i, Z ) is the minimal polynomial of z i+1 over Q(z i ), for each i. Defn. The isomorphism tree I F F is the following subtree of F <ω : { z 1, z 2,..., z m : ( i < m) f i+1 ( z i, z i+1 ) = 0}. Here f i (Y, Z ) is the image of f i (Y, Z ) under the isomorphism of the prime subfields. So I F F is a finite-branching tree, and paths through it correspond to isomorphisms from F onto F, with each path Turing-equivalent to its isomorphism. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 13 / 19

22 Scott Families for Algebraic Fields To enumerate a Scott family Σ for F, we need to give an -formula ψ i for each z i such that, for all z F, ψ i (z) holds in F z i, z B F. For the finitely many roots of f i (z i 1, Z ) in F, ψ i needs to know some level m of I FF such that all nodes at level i with extensions to level m are extendible (i.e. lie on paths). If we have a computable function which gives such a level m for every m, then F has a c.e. Scott family, hence is relatively computably categorical. This function gives a computable bound on the height of the tree I FF above nonextendible nodes. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 14 / 19

23 Back to Computable Categoricity Theorem (HKMS 2010) There exists a computable algebraic field F which is computably categorical, but not relatively c.c. In particular, B F is not Σ 0 2. Proof: a tree construction of F. A node ρ at level 2e has two outcomes: = and =. It tries to ensure that if the structure computed by the e-th Turing program is a field K e isomorphic to F, then some program P ρ computes an isomorphism between them. Each time larger initial fragments of F and K e are found to embed into each other, ρ makes its stronger outcome = eligible. This outcome does not allow lower-priority nodes to do anything until F and K e match up well enough for P ρ to be sure how to build its isomorphism. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 15 / 19

24 Back to Computable Categoricity Theorem (HKMS 2010) There exists a computable algebraic field F which is computably categorical, but not relatively c.c. In particular, B F is not Σ 0 2. Proof: a tree construction of F. A node ρ at level 2e has two outcomes: = and =. It tries to ensure that if the structure computed by the e-th Turing program is a field K e isomorphic to F, then some program P ρ computes an isomorphism between them. Each time larger initial fragments of F and K e are found to embed into each other, ρ makes its stronger outcome = eligible. This outcome does not allow lower-priority nodes to do anything until F and K e match up well enough for P ρ to be sure how to build its isomorphism. Suppose ρ is on the true path. If F = K e, then ρˆ = will also be on the true path, and P ρ will compute an isomorphism. (Finitely much information is needed: ρ, and the last stage at which ρ is initialized.) Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 15 / 19

25 Computable Categoricity, continued Nodes τ at levels 2e + 1 ensure that the e-th partial computable function ϕ e does not compute an m-reduction from B F to. (If this holds for every e, then B F m, hence cannot be Σ 0 2.) τ adds to F two Q-conjugates x τ and y τ. At all stages, there will be two distinct z s and z t already in F such that the minimal polynomials of each over x τ have no root over y τ. Whenever W ϕe( x τ,y τ ) gets a new element, we add a root over y τ of the minimal polynomial of z s over x τ (where s < t), but also add a new u > t for which F has no root over y τ of the minimal polynomial of z u over x τ. Therefore, if τ is never injured, then x τ, y τ B F iff W ϕe( x τ,y τ ) is infinite. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 16 / 19

26 B F and the Galois Group of F over L Extend the definition to field extensions F/L with F computable: B F /L = { a; b F 2 : ( σ Gal(F/L)) σ(a) = b}. For algebraic F/L, if L is a c.e. subfield of F, B F /L is always Π 0 2. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 17 / 19

27 B F and the Galois Group of F over L Extend the definition to field extensions F/L with F computable: B F /L = { a; b F 2 : ( σ Gal(F/L)) σ(a) = b}. For algebraic F/L, if L is a c.e. subfield of F, B F /L is always Π 0 2. Let B F /L be c.e. Then given a, b B F /L, we can compute some σ Gal(F/L) with σ(a) = b. Let F = {x 0, x 1,...}, and let σ(x s ) be the first x t with a, x 0,..., x s ; b, σ(x 0 ),..., σ(x s 1 ), x t A F /L. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 17 / 19

28 B F and the Galois Group of F over L Extend the definition to field extensions F/L with F computable: B F /L = { a; b F 2 : ( σ Gal(F/L)) σ(a) = b}. For algebraic F/L, if L is a c.e. subfield of F, B F /L is always Π 0 2. Let B F /L be c.e. Then given a, b B F /L, we can compute some σ Gal(F/L) with σ(a) = b. Let F = {x 0, x 1,...}, and let σ(x s ) be the first x t with a, x 0,..., x s ; b, σ(x 0 ),..., σ(x s 1 ), x t A F /L. We suggest: Definition A computable algebraic extension F/L has computably approximable Galois group Gal(F/L) if B F /L is computably enumerable. Gal(F/L) is essentially a type-2 computable object, in the sense of computable analysis. (It may have 2 ω -many elements!) Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 17 / 19

29 Automorphism Groups in General Defn. For any structure M with domain ω in which all orbits are finite, say that Aut(M) is d-computably approximable if the following set is computably enumerable in the Turing degree d: A M = { a; b ( ω 2n) : ( σ Aut(M))( i < n)σ(a i ) = b i }. n In general A M is Σ 1 1. For relatively computably categorical structures M, Aut(M) is M-computably approximable: enumerate a; b into A M whenever some ψ in a (computably enumerable) Scott family for M is found to be satisfied by both a and b. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 18 / 19

30 Automorphism Groups in General Defn. For any structure M with domain ω in which all orbits are finite, say that Aut(M) is d-computably approximable if the following set is computably enumerable in the Turing degree d: A M = { a; b ( ω 2n) : ( σ Aut(M))( i < n)σ(a i ) = b i }. n In general A M is Σ 1 1. For relatively computably categorical structures M, Aut(M) is M-computably approximable: enumerate a; b into A M whenever some ψ in a (computably enumerable) Scott family for M is found to be satisfied by both a and b. However, Steiner has found computable structures M with all orbits finite and A M computable, such that M is not computably categorical. For instance, let M be an equivalence relation with exactly one equivalence class of each finite size. We saw the same above for a computable algebraic field. Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 18 / 19

31 Standard References on Computable Fields A. Frohlich & J.C. Shepherdson; Effective procedures in field theory, Phil. Trans. Royal Soc. London, Series A 248 (1956) 950, M. Rabin; Computable algebra, general theory, and theory of computable fields, Transactions of the American Mathematical Society 95 (1960), Yu.L. Ershov; Theorie der Numerierungen III, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 23 (1977) 4, G. Metakides & A. Nerode; Effective content of field theory, Annals of Mathematical Logic 17 (1979), M.D. Fried & M. Jarden, Field Arithmetic (Berlin: Springer-Verlag, 1986). V. Stoltenberg-Hansen & J.V. Tucker; Computable rings and fields, in Handbook of Computability Theory, ed. E.R. Griffor (Amsterdam: Elsevier, 1999), Miller & Shlapentokh (CUNY & ECU) Fields and Computable Categoricity GWU Logic Seminar 19 / 19