Computable Fields and their Algebraic Closures Russell Miller - PowerPoint PPT Presentation

Computable Fields and their Algebraic Closures Russell Miller Queens College & CUNY Graduate Center New York, NY. Workshop on Computability Theory Universidade dos Ac ¸ores Ponta Delgada, Portugal, 6 July 2010 Slides available at qc.edu/˜rmiller/slides.html Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 1 / 17

Classical Algebraic Closures Theorem Every field F has an algebraic closure F : a field extension of F which is algebraically closed and algebraic over F . This algebraic closure of F is unique up to F -isomorphism. Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 2 / 17

Classical Algebraic Closures Theorem Every field F has an algebraic closure F : a field extension of F which is algebraically closed and algebraic over F . This algebraic closure of F is unique up to F -isomorphism. The theory Th ( ACF m ) of algebraically closed fields of characteristic m is κ -categorical for every uncountable κ , and has countable models F m ≺ F m ( X 0 ) ≺ F m ( X 0 , X 1 ) ≺ · · · ≺ F m ( X 0 , X 1 , X 2 , . . . ) . So ACF’s of characteristic m are indexed by their transcendence degrees. (Here F 0 = Q and F p = Z / ( p Z ) for prime p .) Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 2 / 17

Classical Algebraic Closures Theorem Every field F has an algebraic closure F : a field extension of F which is algebraically closed and algebraic over F . This algebraic closure of F is unique up to F -isomorphism. The theory Th ( ACF m ) of algebraically closed fields of characteristic m is κ -categorical for every uncountable κ , and has countable models F m ≺ F m ( X 0 ) ≺ F m ( X 0 , X 1 ) ≺ · · · ≺ F m ( X 0 , X 1 , X 2 , . . . ) . So ACF’s of characteristic m are indexed by their transcendence degrees. (Here F 0 = Q and F p = Z / ( p Z ) for prime p .) Fact All countable ACF’s are computably presentable. Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 2 / 17

Splitting Algorithms Theorem (Kronecker, 1882) The field Q has a splitting algorithm: it is decidable which polynomials in Q [ X ] have factorizations in Q [ X ] . Let F be a computable field of characteristic 0 with a splitting algorithm. Every primitive extension F ( x ) of F also has a splitting algorithm, which may be found uniformly in the minimal polynomial of x over F (or uniformly knowing that x is transcendental over F ). Recall that for x ∈ E algebraic over F , the minimal polynomial of x over F is the unique monic irreducible p ( X ) ∈ F [ X ] with p ( x ) = 0. Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 3 / 17

Splitting Algorithms Theorem (Kronecker, 1882) The field Q has a splitting algorithm: it is decidable which polynomials in Q [ X ] have factorizations in Q [ X ] . Let F be a computable field of characteristic 0 with a splitting algorithm. Every primitive extension F ( x ) of F also has a splitting algorithm, which may be found uniformly in the minimal polynomial of x over F (or uniformly knowing that x is transcendental over F ). Recall that for x ∈ E algebraic over F , the minimal polynomial of x over F is the unique monic irreducible p ( X ) ∈ F [ X ] with p ( x ) = 0. Corollary For any algebraic computable field F , every finitely generated subfield Q ( x 1 , . . . , x n ) or F p ( x 1 , . . . , x n ) has a splitting algorithm, uniformly in the tuple � x 1 , . . . , x d � . Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 3 / 17

Computable Algebraic Closures We want a presentation of F with F as a recognizable subfield. Defn. For a computable field F , a Rabin embedding of F consists of a computable field E and a field homomorphism g : F → E such that: E is algebraically closed; E is algebraic over the image g ( F ) ; and g is a computable function. Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 4 / 17

Computable Algebraic Closures We want a presentation of F with F as a recognizable subfield. Defn. For a computable field F , a Rabin embedding of F consists of a computable field E and a field homomorphism g : F → E such that: E is algebraically closed; E is algebraic over the image g ( F ) ; and g is a computable function. Rabin’s Theorem (1960); see also Frohlich & Shepherdson (1956) Every computable field F has a Rabin embedding. Moreover, for every Rabin embedding g : F → E , the following are Turing-equivalent: the image g ( F ) , as a subset of E ; the splitting set S F = { p ∈ F [ X ] : p factors nontrivially in F [ X ] } ; the root set R F = { p ∈ F [ X ] : p has a root in F } . Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 4 / 17

Proof of Rabin’s Theorem R F ≤ T S F Given p ( X ) , an S F -oracle allows us to find the irreducible factors of p in F [ X ] . But p ∈ R F iff p has a linear factor. Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 5 / 17

Proof of Rabin’s Theorem R F ≤ T S F Given p ( X ) , an S F -oracle allows us to find the irreducible factors of p in F [ X ] . But p ∈ R F iff p has a linear factor. S F ≤ T g ( F ) Given a monic p ( X ) ∈ F [ X ] , find all its roots r 1 , . . . , r d ∈ E . Factorizations of its image p g in E [ X ] are all of the form p g ( X ) = h ( X ) · j ( X ) = (Π i ∈ S ( X − r i )) · (Π i / ∈ S ( X − r i )) for some S � { 1 , . . . , d } . Check if any of these factors lies in g ( F )[ X ] . Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 5 / 17

Proof of Rabin’s Theorem R F ≤ T S F Given p ( X ) , an S F -oracle allows us to find the irreducible factors of p in F [ X ] . But p ∈ R F iff p has a linear factor. S F ≤ T g ( F ) Given a monic p ( X ) ∈ F [ X ] , find all its roots r 1 , . . . , r d ∈ E . Factorizations of its image p g in E [ X ] are all of the form p g ( X ) = h ( X ) · j ( X ) = (Π i ∈ S ( X − r i )) · (Π i / ∈ S ( X − r i )) for some S � { 1 , . . . , d } . Check if any of these factors lies in g ( F )[ X ] . g ( F ) ≤ T R F Given x ∈ E , find some p ( X ) ∈ F [ X ] for which p g ( x ) = 0. Find all roots of p in F : if p ∈ R F , find a root r 1 ∈ F , then check if p ( X ) X − r 1 ∈ R F , etc. Then x ∈ g ( F ) iff x is the image of one of these roots. Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 5 / 17

Different Presentations of F Theorem Let F ∼ = ˜ F be two computable presentations of the same field. Assume that F is algebraic (over its prime subfield Q or F p ). Then R F ≡ T R ˜ F . Proof: Given p ( X ) ∈ F [ X ] , find q ( X ) ∈ F m [ X ] divisible by p ( X ) . Use F to find all roots of h ( q )( X ) in ˜ R ˜ F . Then find the same number of roots of q ( X ) in F , and check whether any one is a root of p ( X ) . ∼ ˜ F = F � | � | ˜ h : → F m F m Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 6 / 17

Comparing R F , S F , and g ( F ) We know that R F ≡ T S F ≡ T g ( F ) . Is there any way to distinguish the complexity of these sets? Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 7 / 17

Comparing R F , S F , and g ( F ) We know that R F ≡ T S F ≡ T g ( F ) . Is there any way to distinguish the complexity of these sets? Recall: A ≤ 1 B if there is a 1-1 computable f such that: ( ∀ x )[ x ∈ A ⇐ ⇒ f ( x ) ∈ B ] . A ≤ wtt B if there are Φ e and a computable bound g with: ( ∀ x )Φ B ↾ g ( x ) ( x ) ↓ = χ A ( x ) . e Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 7 / 17

Comparing R F , S F , and g ( F ) We know that R F ≡ T S F ≡ T g ( F ) . Is there any way to distinguish the complexity of these sets? Recall: A ≤ 1 B if there is a 1-1 computable f such that: ( ∀ x )[ x ∈ A ⇐ ⇒ f ( x ) ∈ B ] . A ≤ wtt B if there are Φ e and a computable bound g with: ( ∀ x )Φ B ↾ g ( x ) ( x ) ↓ = χ A ( x ) . e Theorem (M, 2010) For all algebraic computable fields F , S F ≤ 1 R F . However, there exists such a field F with R F �≤ 1 S F . Problem: Given a polynomial p ( X ) ∈ F [ X ] , compute another polynomial q ( X ) ∈ F [ X ] such that p ( X ) splits ⇐ ⇒ q ( X ) has a root . Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 7 / 17

p ( X ) splits ⇐ ⇒ q ( X ) has a root . Let F t be the subfield F m [ a 0 , . . . , a t − 1 ] . So every F t has a splitting algorithm. For a given p ( X ) , find an t with p ∈ F t [ X ] . Check first whether p splits there. If so, pick its q ( X ) to be a linear polynomial. If not, find the splitting field K t of p ( X ) over F t , and the roots r 1 , . . . , r d of p ( X ) in K t . Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 8 / 17

p ( X ) splits ⇐ ⇒ q ( X ) has a root . Let F t be the subfield F m [ a 0 , . . . , a t − 1 ] . So every F t has a splitting algorithm. For a given p ( X ) , find an t with p ∈ F t [ X ] . Check first whether p splits there. If so, pick its q ( X ) to be a linear polynomial. If not, find the splitting field K t of p ( X ) over F t , and the roots r 1 , . . . , r d of p ( X ) in K t . Proposition For F t ⊆ L ⊆ K t , p ( X ) splits in L [ X ] iff there exists ∅ � S � { r 1 , . . . , r d } such that L contains all elementary symmetric polynomials in S . Russell Miller (CUNY) Algebraic Closures of Computable Fields WCT 2010 Azores 8 / 17