PPaulo Oliva (Ed.): Classical Logic and Computation 2014EPTCS 164, 2014, pp. 18–32, doi:10.4204/EPTCS.164.2 c (cid:13)
B. Mannaa & T. CoquandThis work is licensed under theCreative Commons Attribution License.
A Sheaf Model of the Algebraic Closure
Bassel Mannaa Thierry Coquand
Department of Computer Science and EngineeringUniversity of Gothenburg ∗ Gothenburg, Sweden [email protected] [email protected]
In constructive algebra one cannot in general decide the irreducibility of a polynomial over a field K .This poses some problems to showing the existence of the algebraic closure of K . We give a possibleconstructive interpretation of the existence of the algebraic closure of a field in characteristic 0 bybuilding, in a constructive metatheory, a suitable site model where there is such an algebraic closure.One can then extract computational content from this model. We give examples of computation basedon this model. Since in general it is not decidable whether a given polynomial over a field is irreducible, even whenthe field is given explicitly [6], the notion of algebraic field extension and consequently the notion ofalgebraic closure becomes problematic from a constructive point of view. Even in situations where onecan constructively assert the existence of an algebraic closure of a field [14, Ch. 6] the computationalcontent of such assertions are not always clear. We present a constructive interpretation of the algebraicclosure of field K in characteristic 0 as a site model. Our approach is different from [15] in that we do notassume a polynomial over a field to be decomposable into irreducible factors. The model presented herehas a direct computational content and can be viewed as a model of dynamical evaluation in the sense ofDuval [5] (see also [4]). The site, described in section 3, is given by the category of finitely presented(von Neumann) regular algebras over K with the appropriate Grothendieck topology. In section 4 weprove that the topos E of sheaves on this site contains a model of an algebraically closed field extensionof K . An alternative approach using profinite Galois group is presented in [8]. We also investigate someof the properties of the topos E . Theorem 6.3 shows that the axiom of choice fails to hold in E whenever K is not algebraically closed. Theorem 6.4 shows that when the base field K is the rationals the weakeraxiom of dependent choice fails to hold. We restrict ourselves to constructive metatheory throughoutthe paper with the exception of section 8 in which we show that in a classical metatheory the topos E is boolean (Theorem 8.6). As we will demonstrate by Theorem 8.8 this cannot be shown to hold in anintuitionistic metatheory. In this section we recall some notions that we will use in the remainder the paper, mostly followingthe presentation in [7]. A coverage on a category C is a function J assigning to each object C of C acollection J ( C ) of families of morphisms with codomain C such that for any { f i : C i → C } i ∈ I ∈ J ( C ) andmorphism g : D → C of C there exist { h j : D j → D } j ∈ J ∈ J ( D ) such that for each j ∈ J the morphism ∗ The research leading to this work has been supported by ERC Advanced grant project 247219. . Mannaa & T. Coquand gh j factors through f (cid:96) for some (cid:96) ∈ I . A family S ∈ J ( C ) is called elementary cover or elementarycovering family of C . A site is a category with coverage ( C , J ) . For a presheaf P : C op → Set and family S = { g i : A i → A } i ∈ I of morphisms of C we say that a family { s i ∈ P ( A i ) } i ∈ I is compatible if for each (cid:96), j ∈ I whenever we have h : B → A (cid:96) and f : B → A j such that g (cid:96) h = g j f then s (cid:96) h = s j f , where by s (cid:96) h wemean the restriction of s (cid:96) along h , i.e. P ( h ) s (cid:96) . A presheaf P is a sheaf on the site ( C , J ) if for any object C and any { f i : C i → C } i ∈ I ∈ J ( C ) if { s i ∈ P ( C i ) } i ∈ I is compatible then there exist a unique s ∈ P ( C ) suchthat s f i = s i . We call such s the amalgamation of { s i } i ∈ I . Let J be a coverage on C we define a closure J ∗ of J as follows: For all objects C of C i. { C C −→ C } ∈ J ∗ ( C ) , ii. If S ∈ J ( C ) then S ∈ J ∗ ( C ) , and, iii. If { C i f i −→ C } i ∈ I ∈ J ∗ ( C ) and for each i ∈ I , { C i j g ij −→ C i } j ∈ J i ∈ J ∗ ( C i ) then { C i j f i g ij −−→ C } i ∈ I , j ∈ J i ∈ J ∗ ( C ) . Afamily T ∈ J ∗ ( C ) is called cover or covering family of C .We work with a typed language with equality L [ V , ..., V n ] having the basic types V , ..., V n and typeformers − × − , ( − ) − , P ( − ) . The language L [ V , ..., V n ] has typed constants and function symbols. Forany type Y one has a stock of variables y , y , ... of type Y . Terms and formulas of the language aredefined as usual. We work within the proof theory of intuitionistic higher-order logic (IHOL). A detaileddescription of this deduction system is given in [1].The language L [ V , ..., V n ] along with deduction system IHOL can be interpreted in an elementarytopos in what is referred to as topos semantics . For a sheaf topos this interpretation takes a simpler formreminiscent of Beth semantics, usually referred to as Kripke–Joyal sheaf semantics . We describe thissemantics here briefly following [15].Let E = Sh ( C , J ) be a sheaf topos. An interpretation of the language L [ V , ..., V n ] in the topos E is given as follows: Associate to each basic type V i of L [ V , ..., V n ] an object V i of E . If Y and Z aretypes of L [ V , ..., V n ] interpreted by objects Y and Z , respectively, then the types Y × Z , Y Z , P ( Z ) areinterpreted by Y × Z , Y Z , Ω Z , respectively, where Ω is the subobject classifier of E . A constant e of type E is interpreted by an arrow e −→ E where E is the interpretation of E . For a term τ and an object X of E , we write τ : X to mean τ has a type X interpreted by the object X .Let φ ( x , ..., x n ) be a formula with variables x : X , ..., x n : X n . Let c ∈ X j ( C ) , ..., c n ∈ X n ( C ) for someobject C of C . We define the relation C forces φ ( x , ..., x n )[ c , ..., c n ] written C (cid:13) φ ( x , ..., x n )[ c , ..., c n ] by induction on the structure of φ . Definition 2.1 (Forcing) . First we replace the constants in φ by variables of the same type as follows:Let e : E , ..., e m : E m be the constants in φ ( x , ..., x n ) then C (cid:13) φ ( x , ..., x n )[ c , ..., c n ] iff C (cid:13) φ [ y / e , ..., y m / e m ]( y , ..., y m , x , ..., x n )[ e C ( ∗ ) , ..., e m C ( ∗ ) , c , ..., c n ] where y i : E i and e i : → E i is the interpretation of e i .Now it suffices to define the forcing relation for formulas free of constants by induction as follows: (cid:62) C (cid:13) (cid:62) . ⊥ C (cid:13) ⊥ iff the empty family is a cover of C . = C (cid:13) ( x = x )[ c , c ] iff c = c . ∧ C (cid:13) ( φ ∧ ψ )( x , ..., x n )[ c , ..., c n ] iff C (cid:13) φ ( x , ..., x n )[ c , ..., c n ] and C (cid:13) ψ ( x , ..., x n )[ c , ..., c n ] . ∨ C (cid:13) ( φ ∨ ψ )( x , ..., x n )[ c , ..., c n ] iff there exist a cover { C i f i −→ C } i ∈ I ∈ J ∗ ( C ) such that C i (cid:13) φ ( x , ..., x n )[ c f i , ..., c n f i ] or C i (cid:13) ψ ( x , ..., x n )[ c f i , ..., c n f i ] for each i ∈ I . ⇒ C (cid:13) ( φ ⇒ ψ )( x , ..., x n )[ c , ..., c n ] iff for every morphism f : D → C whenever D (cid:13) φ ( x , ..., x n )[ c f , ..., c n f ] one has D (cid:13) ψ ( x , ..., x n )[ c f , ..., c n f ] .0 A Sheaf Model of the Algebraic Closure
Let y be a variable of the type Y interpreted by the object Y of E . ∃ C (cid:13) ( ∃ y φ ( x , ..., x n , y ))[ c , ..., c n ] iff there exist a cover { C i f i −→ C } i ∈ I ∈ J ∗ ( C ) such that for each i ∈ I one has C i (cid:13) φ ( x , ..., x n , y )[ c f i , ..., c n f i , d ] for some d ∈ Y ( C i ) . ∀ C (cid:13) ( ∀ y φ ( x , ..., x n , y ))[ c , ..., c n ] iff for every morphism f : D → C and for all d ∈ Y ( D ) one has D (cid:13) φ ( x , ..., x n , y )[ c f , ..., c n f , d ] .We have the following derivable local character and monotonicity laws:LC If { C i f i −→ C } i ∈ I ∈ J ∗ ( C ) and for all i ∈ I , C i (cid:13) φ ( x , ..., x n )[ c f i , ..., c n f i ] then C (cid:13) φ ( x , ..., x n )[ c , ..., c n ] .M If C (cid:13) φ ( x , ..., x n )[ c , ..., c n ] and f : D → C then D (cid:13) φ ( x , ..., x n )[ c f , ..., c n f ] . Sh ( RA opK , J ) Definition 3.1 (Regular ring) . A commutative ring R is (von Neumann) regular if for every element a ∈ R there exist b ∈ R such that aba = a and bab = b . This element b is called the quasi-inverse of a .The quasi-inverse of an element a is unique for a [9, Ch. 4]. We thus use the notation a ∗ to referto the quasi-inverse of a . A ring is regular iff it is zero-dimensional and reduced. To be regular isequivalent to the fact that any principal ideal (consequently, any finitely generated ideal) is generatedby an idempotent. If R is regular and a ∈ R then e = aa ∗ is an idempotent such that (cid:104) e (cid:105) = (cid:104) a (cid:105) and R isisomorphic to R × R with R = R / (cid:104) e (cid:105) and R = R / (cid:104) − e (cid:105) . Furthermore a is 0 on the component R and invertible on the component R . Definition 3.2 (Fundamental system of orthogonal idempotents) . A family ( e i ) i ∈ I of idempotents in aring R is a fundamental system of orthogonal idempotents if ∑ i ∈ I e i = ∀ i , j [ i (cid:54) = j ⇒ e i e j = ] . Lemma 3.3.
Given a fundamental system of orthogonal idempotents ( e i ) i ∈ I in a ring A we have a de-composition A ∼ = ∏ i ∈ I A / (cid:104) − e i (cid:105) .Proof. Follows by induction from the fact that A ∼ = A / (cid:104) e (cid:105) × A / (cid:104) − e (cid:105) for an idempotent e ∈ A . Definition 3.4 (Separable polynomial) . Let R be a ring. A polynomial p ∈ R [ X ] is separable if there exist r , s ∈ R [ X ] such that r p + sp (cid:48) =
1, where p (cid:48) ∈ R [ X ] is the derivative of p . Definition 3.5.
A ring R is a (strict) B´ezout ring if for all a , b ∈ R we can find g , a , b , c , d ∈ R such that a = a g , b = b g and ca + db = R is a regular ring then R [ X ] is a strict B´ezout ring (and the converse is true [9]). Intuitively we cancompute the gcd as if R was a field, but we may need to split R when deciding if an element is invertibleor 0. Using this, we see that given a , b in R [ X ] we can find a decomposition R , . . . , R n of R and for each i we have g , a , b , c , d in R i [ X ] such that a = a g , b = b g and ca + db = g monic. Lemma 3.6.
If R is regular and p in R [ X ] is a separable polynomial then R [ a ] = R [ X ] / (cid:104) p (cid:105) is regular.Proof. If c = q ( a ) is an element of R [ a ] with q in R [ X ] we compute the gcd g of p and q . If p = gp , wecan find u and v in R [ X ] such that ug + vp = p is separable. We then have g ( a ) p ( a ) = u ( a ) g ( a ) + v ( a ) p ( a ) =
1. It follows that e = u ( a ) g ( a ) is idempotent and we have (cid:104) e (cid:105) = (cid:104) g ( a ) (cid:105) . . Mannaa & T. Coquand A over a field K is finitely presented if it is of the form K [ X , .., X n ] / (cid:104) f , ..., f m (cid:105) , i.e. thequotient of the polynomial ring over K in finitely many variables by a finitely generated ideal.In order to build the classifying topos of a coherent theory T it is customary in the literature to con-sider the category of all finitely presented T algebras where T is an equational subtheory of T . Theaxioms of T then give rise to a coverage on the dual category [11, Ch. 9]. For our purpose considerthe category C of finitely presented K -algebras. Given an object R of C , the axiom schema of alge-braic closure and the field axiom give rise to families (i.) R → R [ X ] / (cid:104) p (cid:105) where p ∈ R [ X ] is monic and(ii.) R / h a i R R [ a ] , for a ∈ R . Dualized, these are elementary covering families of R in C op . Weobserve however that we can limit our consideration only to those finitely presented K -algebras that arezero dimensional and reduced, i.e. regular. In this case we can assume a is an idempotent and we onlyconsider extensions R [ X ] / (cid:104) p (cid:105) where p is separable.Let RA K be the small category of finitely presented regular algebras over a fixed field K and K -homomorphisms. First we fix an countable set of names S . An object of RA K is a regular algebra of theform K [ X , ..., X n ] / (cid:104) f , ..., f m (cid:105) where X i ∈ S for all 1 ≤ i ≤ n . Note that for any object R , there is a uniquemorphism K → R . A finitely presented regular K -algebra A is a finite dimensional K -algebra, i.e. A hasa finite dimension as a vector space over K [9, Ch 4, Theorem 8.16]. The trivial ring 0 is the terminalobject in the category RA K and K is its initial object.To specify a coverage J on the category RA opK , we define for each object A a collection J op ( A ) offamilies of morphisms of RA K with domain A . We then take J ( A ) to be the dual of J op ( A ) in the sensethat { ϕ i : A i → A } i ∈ I ∈ J ( A ) if and only if { ϕ i : A → A i } i ∈ I ∈ J op ( A ) where ϕ i of RA K is the dual of ϕ i of RA opK . We call J op cocoverage. We call an element of J op ( A ) an elementary cocover (cocoveringfamily) of A . We define J ∗ op similarly. We call elements of J ∗ op ( A ) cocovers (cocovering families) of A .By a separable extension of a ring R we mean a ring R [ a ] = R [ X ] / (cid:104) p (cid:105) where p ∈ R [ X ] is non-constant,monic and separable. Definition 3.7 (Topology for RA opK ) . For an object A of RA K the cocovering families are given by:(i.) If ( e i ) i ∈ I is a fundamental system of orthogonal idempotents of A , then { A ϕ i −→ A / (cid:104) − e i (cid:105)} i ∈ I ∈ J op ( A ) where for each i ∈ I , ϕ i is the canonical homomorphism.(ii.) Let A [ a ] be a separable extension of A . We have { A ϑ −→ A [ a ] } ∈ J op ( A ) where ϑ is the canonicalembedding.Note that in particular 3.7.(i.) implies that the trivial algebra 0 is covered by the empty family ofmorphisms since an empty family of elements in this ring form a fundamental system of orthogonalidempotents. Also note that 3.7.(ii.) implies that { A A −−→ A } ∈ J op ( A ) . Lemma 3.8.
The function J of Definition 3.7 is a coverage on RA opK .Proof. Let η : R → A be a morphism of RA K and S ∈ J op ( R ) . We show that there exist an elementarycocover T ∈ J op ( A ) such that for each ϑ ∈ T , ϑ η factors through some ϕ ∈ S . By duality, this implies J is a coverage on RA opK . By case analysis on the clauses of Definition 3.7.2 A Sheaf Model of the Algebraic Closure (i.) If S = { ϕ i : R → R / (cid:104) − e i (cid:105)} i ∈ I , where ( e i ) i ∈ I is a fundamental system of orthogonal idempotentsof R . In A , the family ( η ( e i )) i ∈ I is fundamental system of orthogonal idempotents. We have anelementary cocover { ϑ i : A → A / (cid:104) − η ( e i ) (cid:105)} i ∈ I ∈ J op ( A ) . For each i ∈ I , the homomorphism η induces a K -homomorphism η e i : R / (cid:104) − e i (cid:105) → A / (cid:104) − η ( e i ) (cid:105) where η e i ( r + (cid:104) − e i (cid:105) ) = η ( r ) + (cid:104) − η ( e i ) (cid:105) . Since ϑ i ( η ( r )) = η ( r ) + (cid:104) − η ( e i ) (cid:105) we have that ϑ i η = η e i ϕ i .(ii.) If S = { ϕ : R → R [ r ] } with R [ r ] = R [ X ] / (cid:104) p (cid:105) and p ∈ R [ X ] monic, non-constant, and separable.Since sp + t p (cid:48) =
1, we have η ( s ) η ( p ) + η ( t ) η ( p (cid:48) ) = η ( s ) η ( p ) + η ( t ) η ( p ) (cid:48) =
1. Then q = η ( p ) ∈ A [ X ] is separable. Let A [ a ] = A [ X ] / (cid:104) q (cid:105) . We have an elementary cocover { ϑ : A → A [ a ] } ∈ J op ( A ) where ϑ is the canonical embedding. Let ζ : R [ r ] → A [ a ] be the K -homomorphism such that ζ | R = η and ζ ( r ) = a . For b ∈ R , we have ϑ ( η ( b )) = ζ ( ϕ ( b )) . Lemma 3.9.
Let P : RA K → Set be a presheaf on RA opK such that P ( ) = . Let R be an object of RA K and let ( e i ) i ∈ I be a fundamental system of orthogonal idempotents of R. For each i ∈ I, let R i = R / (cid:104) − e i (cid:105) and let ϕ i : R → R i be the canonical homomorphism. Any family { s i ∈ P ( R i ) } is compatible.Proof. Let B be an object and for some i , j ∈ I let ϑ : R i → B and ζ : R j → B be such that ϑ ϕ i = ζ ϕ j .We will show that P ( ϑ )( s i ) = P ( ζ )( s j ) .(i.) If i = j , then since ϕ i is surjective we have ϑ = ζ and P ( ϑ ) = P ( ζ ) .(ii.) If i (cid:54) = j , then since e i e j = ϕ i ( e i ) = ϕ j ( e j ) = ϕ j ( e i ) = ϕ j ( e i e j ) =
0. But then1 = ϑ ( ) = ϑ ( ϕ i ( e i )) = ζ ( ϕ j ( e i )) = ζ ( ) = B is the trivial algebra 0. By assumption P ( ) =
1, hence P ( ϑ )( s i ) = P ( ζ )( s j ) = ∗ . Corollary 3.10.
Let F be a sheaf on ( RA opK , J ) . Let R be an object of RA K and ( e i ) i ∈ I a fundamentalsystem of orthogonal idempotents of R. Let R i = R / (cid:104) − e i (cid:105) and ϕ i : R → R i be the canonical homomor-phism. The map f : F ( R ) → ∏ i ∈ I F ( R i ) such that f ( s ) = ( F ( ϕ i ) s ) i ∈ I is an isomorphism.Proof. Since F ( ) =
1, by Lemma 3.9 any family { s i ∈ F ( R i ) } i ∈ I is compatible. Since F is a sheaf, thefamily { s i ∈ F ( R i ) } i ∈ I has a unique amalgamation s ∈ F ( R ) with restrictions s ϕ i = s i . The isomorphismis given by f s = ( s ϕ i ) i ∈ I . We can then use the tuple notation ( s i ) i ∈ I to denote the element s in F ( R ) .One say that a polynomial f ∈ R [ X ] has a formal degree n if f can be written as f = a n X n + ... + a which is to express that for any m > n the coefficient of X m is known to be 0. Lemma 3.11.
Let R be a regular ring and p , p ∈ R [ X ] be monic polynomials of degrees n and n respectively. Let R [ a , b ] = R [ X , Y ] / (cid:104) p ( X ) , p ( Y ) (cid:105) . Let q , q ∈ R [ Z ] be of formal degrees m < n andm < n respectively. If q ( a ) = q ( b ) then q = q = r ∈ R.Proof.
The statement follows immediately since the R -basis a i , i > b j , j > Corollary 3.12.
Let R be an object of RA K and p ∈ R [ X ] separable and monic. Let R [ a ] = R [ X ] / (cid:104) p (cid:105) and ϕ : R → R [ a ] the canonical morphism. Let R [ b , c ] = R [ X , Y ] / (cid:104) p ( X ) , p ( Y ) (cid:105) . The commuting diagram . Mannaa & T. Coquand R [ a ] R [ b , c ] R R [ a ] ϑϕ ϕ ζ ϑ | R = ζ | R = R , ϑ ( a ) = b, ζ ( a ) = cis a pushout diagram of RA K . Moreover, ϕ is the equalizer of ζ and ϑ .Proof. Let R [ a ] B ηρ be morphisms of RA K such that ηϕ = ρϕ . Then for all r ∈ R we have η ( r ) = ρ ( r ) . Let γ : R [ b , c ] → B be the homomorphism such that γ ( r ) = η ( r ) = ρ ( r ) for all r ∈ R while γ ( b ) = η ( a ) , γ ( c ) = ρ ( a ) . Then γ is the unique map such that γϑ = η and γζ = ρ .Let A be an object of RA K and let ε : A → R [ a ] be a map such that ζ ε = ϑ ε . By Lemma 3.11 if forsome f ∈ R [ a ] one has ζ ( f ) = ϑ ( f ) then f ∈ R (i.e. f is of degree 0 as a polynomial in a over R ). Thus ε ( A ) ⊂ R and we can factor ε uniquely (since ϕ is injective) as ε = ϕ µ with µ : A → R .Let { ϕ : R → R [ a ] } be a singleton elementary cocover. Since one can form the pushout of ϕ with itself, the compatibility condition on a singleton family { s ∈ F ( R [ a ]) } can be simplified as: Let R R [ a ] A ϕ ηϑ be a pushout diagram. A family { s ∈ F ( R [ a ]) } is compatible if and only if s ϑ = s η . Corollary 3.13.
The coverage J is subcanonical, i.e. all representable presheaves in Set RA K are sheaveson ( RA opK , J ) . We define the presheaf F : RA K → Set to be the forgetful functor. That is, for an object A of RA K , F ( A ) = A and for a morphism ϕ : A → C of RA K , F ( ϕ ) = ϕ . Lemma 4.1. F is a sheaf of sets on the site ( RA opK , J ) Proof.
By case analysis on the clauses of Definition 3.7.(i.) Let { R ϕ i −→ R / (cid:104) − e i (cid:105)} i ∈ I ∈ J op ( R ) , where ( e i ) i ∈ I is fundamental system of orthogonal idempo-tents of R . The presheaf F has the property F ( ) =
1. By Lemma 3.9 a family { a i ∈ R / (cid:104) − e i (cid:105)} i ∈ I is a compatible family. By the isomorphism R ( ϕ i ) i ∈ I −−−→ ∏ i ∈ I R / (cid:104) − e i (cid:105) the element a = ( a i ) i ∈ I ∈ R is the unique element such that ϕ i ( a ) = a i .(ii.) Let { R ϕ −→ R [ a ] } ∈ J op ( R ) where R [ a ] = R [ X ] / (cid:104) p (cid:105) with p ∈ R [ X ] monic, non-constant andseparable polynomial. Let { r ∈ R [ a ] } be a compatible family. Let R R [ a ] R [ b , c ] ϕ ϑζ bethe pushout diagram of Corollary 3.12. Compatibility then implies ϑ ( r ) = ζ ( r ) which by the sameCorollary is true only if the element r is in R . We then have that r is the unique element restrictingto itself along the embedding ϕ .We fix a field K of characteristic 0. Let L [ F , + , . ] be a language with basic type F and functionsymbols + , . : F × F → F . We extend L [ F , + , . ] by adding a constant symbol of type F for each element a ∈ K , to obtain L [ F , + , . ] K . Define Diag ( K ) as : if φ is an atomic L [ F , + , . ] K -formula or the negation4 A Sheaf Model of the Algebraic Closure of one such that K | = φ ( a , ..., a n ) then φ ( a , ..., a n ) ∈ Diag ( K ) . The theory T equips the type F withaxioms of the geometric theory of algebraically closed field containing K Definition 4.2.
The theory T has the following sentences (with all the variables having the type F ).1. Diag ( K ) .2. The axioms of a commutative group: (a) ∀ x [ + x = x + = x ] (b) ∀ x ∀ y ∀ z [ x + ( y + z ) = ( x + y ) + z ] (c) ∀ x ∃ y [ x + y = ] (d) ∀ x ∀ y [ x + y = y + x ]
3. The axioms of a commutative ring: (a) ∀ x [ x = x ] (b) ∀ x [ x = ] (c) ∀ x ∀ y [ xy = yx ] (d) ∀ x ∀ y ∀ z [ x ( yz ) = ( xy ) z ] (e) ∀ x ∀ y ∀ z [ x ( y + z ) = xy + xz ]
4. The field axioms: (a) 1 (cid:54) =
0. (b) ∀ x [ x = ∨ ∃ y [ xy = ]] .5. The axiom schema for algebraic closure: ∀ a . . . ∀ a n ∃ x [ x n + ∑ ni = x n − i a i = ] .6. F is algebraic over K : ∀ x [ (cid:87) p ∈ K [ Y ] p ( x ) = ] .With these axioms the type F becomes the type of an algebraically closed field containing K . Weproceed to show that with the interpretation of the type F by the object F the topos Sh ( RA opK , J ) is amodel of T , i.e. F is a model, in Kripke–Joyal semantics, of an algebraically closed field containing of K . First note that since there is a unique map K → C for any object C of RA K , an element a ∈ K givesrise to a unique map a −→ F , that is the map ∗ (cid:55)→ a ∈ F ( K ) . Every constant a ∈ K of the language is theninterpreted by the corresponding unique arrow a −→ F . (we use the same symbol for constants and theirinterpretation to avoid cumbersome notation). That F satisfies Diag ( K ) then follows directly. Lemma 4.3. F is a ring object.Proof.
For an object C of RA K the object F ( C ) is a commutative ring. Lemma 4.4. F is a field.Proof.
For any object R of RA K one has R (cid:13) (cid:54) = R ϕ −→ C such that C (cid:13) = C is trivial and thus C (cid:13) ⊥ . Next we show that for variables x and y of type F and any object R of RA opK we have R (cid:13) ∀ x [ x = ∨ ∃ y [ xy = ]] . Let ϕ : A → R be a morphism of RA opK and let a ∈ A . We needto show that A (cid:13) a = ∨ ∃ y [ ya = ] . The element e = aa ∗ is an idempotent and we have a cover { ϕ : A / (cid:104) e (cid:105) → A , ϕ : A / (cid:104) − e (cid:105) → A } ∈ J ∗ ( A ) with A / (cid:104) e (cid:105) (cid:13) a ϕ = A / (cid:104) − e (cid:105) (cid:13) ( a ϕ )( a ∗ ϕ ) = e ϕ = ∃ we have A / (cid:104) − e (cid:105) (cid:13) ∃ y [( a ϕ ) y = ] and by ∨ , A / (cid:104) − e (cid:105) (cid:13) a ϕ = ∨ ∃ y [( a ϕ ) y = ] .Similarly, A / (cid:104) e (cid:105) (cid:13) a ϕ = ∨ ∃ y [( a ϕ ) y = ] . By ∀ we get R (cid:13) ∀ x [ x = ∨ ∃ y [ xy = ]] .To show that A (cid:13) ∀ a . . . ∀ a n ∃ x [ x n + ∑ ni = x n − i a i = ] for every n , we need to be able to extend analgebra R of RA K with the appropriate roots. We need the following lemma. Lemma 4.5.
Let L be a field and f ∈ L [ X ] a monic polynomial. Let g = (cid:104) f , f (cid:48) (cid:105) , where f (cid:48) is the derivativeof f . Writing f = hg we have that h is separable. We call h the separable associate of f .Proof. Let a be the gcd of h and h (cid:48) . We have h = l a . Let d be the gcd of a and a (cid:48) . We have a = l d and a (cid:48) = m d , with l and m coprime.The polynomial a divides h (cid:48) = l a (cid:48) + l (cid:48) a and hence that a = l d divides l a (cid:48) = l m d . It follows that l divides l m and since l and m are coprime, that l divides l .Also, if a n divides p then p = qa n and p (cid:48) = q (cid:48) a n + nqa (cid:48) a n − . Hence da n − divides p (cid:48) . Since l divides l , this implies that a n = l da n − divides l p (cid:48) . So a n + divides al p (cid:48) = hp (cid:48) . . Mannaa & T. Coquand a divides f and f (cid:48) , a divides g . We show that a n divides g for all n by induction on n . If a n divides g we have just seen that a n + divides g (cid:48) h . Also a n + divides h (cid:48) g since a divides h (cid:48) . So a n + divides g (cid:48) h + h (cid:48) g = f (cid:48) . On the other hand, a n + divides f = hg = l ag . So a n + divides g which is thegcd of f and f (cid:48) . This implies that a is a unit.Since F is a field, the previous lemma holds for polynomials over F . This means that for all objects R of RA opK we have R (cid:13) Lemma 4.5. Thus we have the following Corollary.
Corollary 4.6.
Let R be an object of RA K and let f be a monic polynomial of degree n in R [ X ] and f (cid:48) its derivative. There is a cocover { ϕ i : R → R i } i ∈ I ∈ J ∗ op ( R ) and for each R i we have h , g , q , r , s ∈ R i [ X ] such that ϕ i ( f ) = hg, ϕ i ( f (cid:48) ) = qg and rh + sq = . Moreover, h is monic and separable. Note that in characteristic 0, if f is monic and non-constant the separable associate of f is non-constant. Lemma 4.7.
The field object F ∈ Sh ( RA opK , J ) is algebraically closed.Proof. We prove that for all n > ( a , ..., a n ) ∈ F n ( R ) = R n , one has R (cid:13) ∃ x [ x n + ∑ ni = x n − i a i = ] .Let f = x n + ∑ ni = x n − i a i . By Corollary 4.6 we have a cover { ϑ j : R j → R } j ∈ I ∈ J ∗ ( R ) such that in each R j we have g = (cid:104) f ϑ j , f (cid:48) ϑ j (cid:105) and f ϑ j = hg with h ∈ R j [ X ] monic and separable. Note that if deg f ≥ h isnon-constant. For each R j we have a singleton cover { ϕ : R j [ b ] → R j | R j [ b ] = R j [ X ] / (cid:104) h (cid:105)} ∈ J ∗ ( R j ) . Thatis, we have R j [ b ] (cid:13) b n + ∑ ni = b n − ( a i ϑ j ϕ ) =
0. By ∃ we get R j [ b ] (cid:13) ∃ x [ x n + ∑ ni = x n − ( a i ϑ j ϕ ) = ] and by LC we have R j (cid:13) ∃ x [ x n + ∑ ni = x n − ( a i ϑ j ) = ] . Since this is true for each R j , j ∈ J we have byLC R (cid:13) ∃ x [ x n + ∑ ni = x n − a i = ] . Lemma 4.8. F is algebraic over K.Proof.
We will show that for any object R of RA K and element r ∈ R one has R (cid:13) (cid:87) p ∈ K [ X ] p ( r ) = R is a finitely presented K -algebra we have that R is a finite integral extension of a polynomial ring K [ Y , ..., Y n ] ⊂ R where Y , .., Y n are elements of R algebraically independent over K and that R has Krulldimension n [9, Ch 13, Theorem 5.4]. Since R is zero-dimensional (i.e. has Krull dimension 0) we have n = R is integral over K , i.e. any element r ∈ R is the zero of some monic polynomial over K . Here we describe the object of natural numbers in the topos Sh ( RA opK , J ) and the object of power seriesover the field F . This will be used in section 6 to show that the axiom of dependent choice does not holdwhen the base field K is the rationals and later in the example of Newton–Puiseux theorem (section 7).Let P : RA K → Set be a constant presheaf associating to each object A of RA K a discrete set B . Thatis, P ( A ) = B and P ( A ϕ −→ R ) = B for all objects A and all morphism ϕ of RA K . Let (cid:101) P : RA K → Set bethe presheaf such that (cid:101) P ( A ) is the set of elements of the form { ( e i , b i ) } i ∈ I where ( e i ) i ∈ I is a fundamentalsystem of orthogonal idempotents of A and for each i , b i ∈ B . We express such an element as a formal sum ∑ i ∈ I e i b i . Let ϕ : A → R be a morphism of RA K , the restriction of ∑ i ∈ I e i b i ∈ (cid:101) P ( A ) along ϕ is given by ( ∑ i ∈ I e i b i ) ϕ = ∑ i ∈ I ϕ ( e i ) b i ∈ (cid:101) P ( R ) . In particular with canonical morphisms ϕ i : A → A / (cid:104) − e i (cid:105) , one hasfor any j ∈ I that ( ∑ i ∈ I e i b i ) ϕ j = b j ∈ (cid:101) P ( A / (cid:104) − e j (cid:105) ) . Two elements ∑ i ∈ I e i b i ∈ (cid:101) P ( A ) and ∑ j ∈ J d j c j ∈ (cid:101) P ( A ) are equal if and only if ∀ i ∈ I , j ∈ J [ b i (cid:54) = c j ⇒ e i d j = ] .To prove that (cid:101) P is a sheaf we will need the following lemmas.6 A Sheaf Model of the Algebraic Closure
Lemma 5.1.
Let R be a regular ring and let ( e i ) i ∈ I be a fundamental system of orthogonal idempotentsof R. Let R i = R / (cid:104) − e i (cid:105) and ([ d j ]) j ∈ J i be a fundamental system of orthogonal idempotents of R i , where [ d j ] = d j + (cid:104) − e i (cid:105) . The family ( e i d j ) i ∈ I , j ∈ J i is a fundamental system of orthogonal idempotents of R.Proof. In R one has ∑ j ∈ J i e i d j = e i ∑ j ∈ J i d j = e i ( + (cid:104) − e i (cid:105) ) = e i . Hence, ∑ i ∈ I , j ∈ J i e i d j = ∑ i ∈ I e i =
1. Forsome i ∈ I and t , k ∈ J i we have ( e i d t )( e i d k ) = e i ( + (cid:104) − e i (cid:105) ) = R . Thus for i , (cid:96) ∈ I , j ∈ J i and s ∈ J (cid:96) one has i (cid:54) = (cid:96) ∨ j (cid:54) = s ⇒ ( e i d j )( e (cid:96) d s ) = Lemma 5.2.
Let R be a regular ring, f ∈ R [ Z ] a polynomial of formal degree n and p ∈ R [ Z ] a monicpolynomial of degree m > n. If in R [ X , Y ] one has f ( Y )( − f ( X )) = (cid:104) p ( X ) , p ( Y ) (cid:105) then f = e ∈ Rwith e an idempotent.Proof.
Let f ( Z ) = ∑ ni = r i Z i . By the assumption, for some q , g ∈ R [ X , Y ] f ( Y )( − f ( X )) = n ∑ i = r i ( − n ∑ j = r j X j ) Y i = qp ( X ) + gp ( Y ) One has ∑ ni = r i ( − ∑ nj = r j X j ) Y i = g ( X , Y ) p ( Y ) mod (cid:104) p ( X ) (cid:105) . Since p ( Y ) is monic of Y -degree greaterthan n , one has that r i ( − ∑ nj = r j X j ) = (cid:104) p ( X ) (cid:105) for all 0 ≤ i ≤ n . But this means that r i r n X n + r i r n − X n − + ... + r i r − r i is divisible by p ( X ) for all 0 ≤ i ≤ n which because p ( X ) is monic of degree m > n implies that all coefficients are equal to 0. In particular, for 1 ≤ i ≤ n one gets that r i = r i = R is reduced. For i = r r − r = r is an idempotent of R . Lemma 5.3.
The presheaf (cid:101) P described above is a sheaf on ( RA opK , J ) .Proof. By case analysis on Definition 3.7.(i.) Let { R ϕ i −→ R / (cid:104) − e i (cid:105)} i ∈ I ∈ J op ( R ) where ( e i ) i ∈ I be a fundamental system of orthogonal idem-potents of an object R . Let R / (cid:104) − e i (cid:105) = R i . Since (cid:101) P ( ) = { s i ∈ (cid:101) P ( R i ) } i ∈ I is compatible. For each i , Let s i = ∑ j ∈ J i [ d j ] b j . By Lemma 5.1 we have an element s = ∑ i ∈ I , j ∈ J i ( e i d j ) b j ∈ (cid:101) P ( R ) the restriction of which along ϕ i is the element ∑ j ∈ J i [ d j ] b j ∈ (cid:101) P ( R i ) .It remains to show that this is the only such element. Let there be an element ∑ (cid:96) ∈ L c (cid:96) a (cid:96) ∈ (cid:101) P ( R ) that restricts to u i = s i along ϕ i . We have u i = ∑ (cid:96) ∈ L [ c (cid:96) ] a (cid:96) . One has that for any j ∈ J i and (cid:96) ∈ L , b j (cid:54) = a (cid:96) ⇒ [ c (cid:96) d j ] = R i , hence, in R one has b j (cid:54) = a (cid:96) ⇒ c (cid:96) d j = r ( − e i ) . Multiplying both sidesof c (cid:96) d j = r ( − e i ) by e i we get b j (cid:54) = a (cid:96) ⇒ c (cid:96) ( e i d j ) =
0. Thus proving s = ∑ (cid:96) ∈ L c (cid:96) a (cid:96) .(ii.) Let { ϕ : R → R [ a ] = R [ X ] / (cid:104) p (cid:105)} ∈ J op ( R ) where p ∈ R [ X ] is monic non-constant and sepa-rable. Let the singleton { s = ∑ i ∈ I e i b i ∈ (cid:101) P ( R [ a ]) } be compatible. We can assume w.l.o.g. that ∀ i , j ∈ I [ i (cid:54) = j ⇒ b i (cid:54) = b j ] since if b k = b (cid:96) one has that ( e k + e (cid:96) ) b l + ∑ j (cid:54) = (cid:96), j (cid:54) = kj ∈ I e j b j = s . (Note thatan idempotent e i of R [ a ] is a polynomial e i ( a ) in a of formal degree less than deg p ). Let R [ c , d ] = R [ X , Y ] / (cid:104) p ( X ) , p ( Y ) (cid:105) , by Corollary 3.12, one has a pushout diagram R R [ a ] R [ c , d ] ϕ ζϑ where ζ | R = ϑ | R = R , ζ ( a ) = d and ϑ ( a ) = c . That the singleton { s } is compatible thenmeans s ϑ = ∑ i ∈ I e i ( c ) b i = s ζ = ∑ i ∈ I e i ( d ) b i , i.e. ∀ i , j ∈ I [ b i (cid:54) = b j ⇒ e i ( c ) e j ( d ) = ] . By theassumption that b i (cid:54) = b j whenever i (cid:54) = j we have in R [ c , d ] that e j ( d ) e i ( c ) = i (cid:54) = j ∈ I . Thus e j ( d ) ∑ i (cid:54) = j e i ( c ) = e j ( d )( − e j ( c )) =
0, i.e. in R [ X , Y ] one has e j ( Y )( − e j ( X )) = (cid:104) p ( X ) , p ( Y ) (cid:105) . By Lemma 5.2 we have that e j ( X ) = e j ( Y ) = e ∈ R . We have thus shown s is . Mannaa & T. Coquand ∑ j ∈ J d j b j ∈ (cid:101) P ( R [ a ]) such that d j ∈ R for j ∈ J . That is ∑ j ∈ J d j b j ∈ (cid:101) P ( R ) . Thus we havefound a unique (since (cid:101) P ( ϕ ) is injective) element in (cid:101) P ( R ) restricting to s along ϕ . Lemma 5.4.
Let P and (cid:101) P be as described above. Let Γ : P → (cid:101) P be the presheaf morphism such that Γ R ( b ) = b ∈ (cid:101) P ( R ) for any object R and b ∈ B. If E is a sheaf and Λ : P → E is a morphism of presheaves,then there exist a unique sheaf morphism ∆ : (cid:101) P → E such that the following diagram, of Set RA K , com-mutes. P E e P ΛΓ ∆
That is to say, Γ : P → (cid:101) P is the sheafification of P .Proof. Let a = ∑ i ∈ I e i b i ∈ (cid:101) P ( A ) and let A i = A / (cid:104) − e i (cid:105) with canonical morphisms ϕ i : A → A i .Let E and Λ be as in the statement of the lemma. If there exist a sheaf morphism ∆ : (cid:101) P → E ,then ∆ being a natural transformation forces us to have for all i ∈ I , E ( ϕ i ) ∆ A = ∆ A i (cid:101) P ( ϕ i ) . By Lemma3.10, we know that the map d ∈ E ( A ) (cid:55)→ ( E ( ϕ i ) d ∈ E ( A i )) i ∈ I is an isomorphism. Thus it must be that ∆ A ( a ) = ( ∆ A i (cid:101) P ( ϕ i )( a )) i ∈ I = ( ∆ A i ( b i )) i ∈ I . But ∆ A i ( b i ) = ∆ A i Γ A i ( b i ) . To have ∆Γ = Λ we must have ∆ A i ( b i ) = Λ A i ( b i ) . Hence, we are forced to have ∆ A ( a ) = ( Λ A i ( b i )) i ∈ I . Note that ∆ is unique since itsvalue ∆ A ( a ) at any A and a is forced by the commuting diagram above.The constant presheaf of natural numbers N is the natural numbers object in Set RA K . We associateto N a sheaf (cid:101) N as described above. From Lemma 5.4 one can easily show that (cid:101) N satisfy the axioms of anatural numbers object in Sh ( RA opK , J ) . Definition 5.5.
Let F [[ X ]] be the presheaf mapping each object R of RA K to F [[ X ]]( R ) = R [[ X ]] = R N with the obvious restriction maps. Lemma 5.6. F [[ X ]] is a sheaf.Proof. The proof is immediate as a corollary of Lemma 4.1.
Lemma 5.7.
The sheaf F [[ X ]] is naturally isomorphic to the sheaf F (cid:101) N .Proof. Let C be an object of RA opK . Since F (cid:101) N ( C ) ∼ = y C × (cid:101) N → F , an element α C ∈ F (cid:101) N ( C ) is a familyof elements of the form α C , D : y C ( D ) × (cid:101) N ( D ) → F ( D ) where D is an object of RA opK . Define Θ : F (cid:101) N → F [[ X ]] as ( Θ α ) C ( n ) = α C , C ( C , n ) . Define Λ : F [[ X ]] → F (cid:101) N as ( Λ β ) C , D ( C ϕ −→ D , ∑ i ∈ I e i n i ) = ( ϑ i ϕ ( β C ( n i ))) i ∈ I ∈ F ( D ) where D ϑ i −→ D / (cid:104) − e i (cid:105) is the canonical morphism. Note that by Lemma 3.10 one indeed has that ( ϑ i ϕ ( β C ( n i ))) i ∈ I ∈ ∏ i ∈ I F ( D i ) ∼ = F ( D ) . One can easily verify that Θ and Λ are natural. It remains toshow the isomorphism. One one hand we have ( ΛΘ α ) C , D ( ϕ , ∑ i ∈ I e i n i ) = ( ϑ i ϕ (( Θ α ) C ( n i ))) i ∈ I = ( ϑ i ϕ ( α C , C ( C , n i ))) i ∈ I = (( α C , D i ( ϑ i ϕ , n i ))) i ∈ I = α C , D ( ϕ , ∑ i ∈ I e i n i ) Thus showing ΛΘ = F (cid:101) N . On the other hand, ( ΘΛ β ) C ( n ) = ( Λ β ) C , C ( C , n ) = C C ( β C ( n )) = β C ( n ) .Thus ΘΛ = F [[ X ]] .8 A Sheaf Model of the Algebraic Closure
Lemma 5.8.
The power series object F [[ X ]] is a ring object.Proof. A Corollary to Lemma 4.3.
The ( external ) axiom of choice fails to hold (even in a classical metatheory) in the topos Sh ( RA opK , J ) whenever the field K is not algebraically closed. To show this we will show that there is an epimorphismin Sh ( RA opK , J ) with no section. Fact 6.1.
Let Θ : P → G be a morphism of sheaves on a site ( C , J ) . Then Θ is an epimorphism if foreach object C of C and each element c ∈ G ( C ) there is a cover S of C such that for all f : D → C in thecover S the element c f is in the image of Θ D . [10, Ch. 3]. Lemma 6.2.
Let K be a field of characteristic not algebraically closed. There is an epimorphism in Sh ( RA opK , J ) with no section.Proof. Let f = X n + ∑ ni = r i X n − i be a non-constant polynomial for which no root in K exist. w.l.o.g.we assume f separable. One can construct Λ : F → F defined by Λ C ( c ) = c n + ∑ n − i = r i c n − i ∈ C . Given d ∈ F ( C ) , let g = X n + ∑ n − i = r i X n − i − d . By Corollary 4.6 there is a cover { C (cid:96) ϕ (cid:96) −−→ C } (cid:96) ∈ L ∈ J ∗ ( C ) with h (cid:96) ∈ C (cid:96) [ X ] a separable non-constant polynomial dividing g . Let C (cid:96) [ x (cid:96) ] = C (cid:96) [ X ] / (cid:104) h (cid:96) (cid:105) one has a singletoncover { C (cid:96) [ x (cid:96) ] ϑ (cid:96) −→ C (cid:96) } and thus a composite cover { C (cid:96) [ x (cid:96) ] ϑ (cid:96) ϕ (cid:96) −−→ C } (cid:96) ∈ L ∈ J ∗ ( C ) . Since x (cid:96) is a root of h (cid:96) | g we have Λ C (cid:96) [ x (cid:96) ] ( x (cid:96) ) = x n (cid:96) + ∑ n − i = r i x n − i (cid:96) = d or more precisely Λ C (cid:96) [ x (cid:96) ] ( x (cid:96) ) = d ϕ (cid:96) ϑ (cid:96) . Thus, Λ is anepimorphism (by Fact 6.1) and it has no section, for if it had a section Ψ : F → F then one would have Ψ K ( − r n ) = a ∈ K such that a n + ∑ ni = r i a n − i = Theorem 6.3.
Let K be a field of characteristic not algebraically closed. The axiom of choice fails tohold in the topos Sh ( RA opK , J ) . We note that in Per Martin-L¨of type theory one can show that (see [13]) ( ∏ x ∈ A )( ∑ y ∈ B [ x ]) C [ x , y ] ⇒ ( ∑ f ∈ ( ∏ x ∈ A ) B [ x ])( ∏ x ∈ A ) C [ x , f ( x )] As demonstrated in the topos Sh ( RA opK , J ) we have an example of an intuitionistically valid formula ofthe form ∀ x ∃ y φ ( x , y ) where no function f exist for which ∃ f ∀ x φ ( x , f ( x )) holds.We demonstrate further that when the base field is Q the weaker axiom of dependent choice does nothold (internally) in the topos Sh ( RA op Q , J ) . For a relation R ⊂ Y × Y the axiom of dependent choice isstated as ∀ x ∃ yR ( x , y ) ⇒ ∀ x ∃ g ∈ Y N [ g ( ) = x ∧ ∀ nR ( g ( n ) , g ( n + ))] (ADC) Theorem 6.4. Sh ( RA op Q , J ) (cid:13) ¬ ADC .Proof.
Consider the binary relation on the algebraically closed object F defined by the characteristicfunction φ ( x , y ) : = y − x =
0. Assume C (cid:13) ADC for some object C of RA K . Since C (cid:13) ∀ x ∃ y [ y − x = ] we have C (cid:13) ∀ x ∃ g ∈ F (cid:101) N [ g ( ) = x ∧ ∀ n [ g ( n ) = g ( n + )]] . That is for all morphisms C ζ −→ A of RA K and elements a ∈ F ( A ) one has A (cid:13) ∃ g ∈ F (cid:101) N [ g ( ) = a ∧ ∀ n [ g ( n ) = g ( n + )]] . Taking a = A (cid:13) ∃ g ∈ F (cid:101) N [ g ( ) = ∧ ∀ n [ g ( n ) = g ( n + )]] . Which by ∃ implies the existence of a cocover { η i : A → A i | i ∈ I } and power series α i ∈ F (cid:101) N ( A i ) such that A i (cid:13) α i ( ) = ∧ ∀ n [ α i ( n ) = α i ( n + )]] . . Mannaa & T. Coquand F (cid:101) N ( A i ) ∼ = A i [[ X ]] and thus the above forcing implies the existence of a series α i = + / + ... + / j + ... ∈ A i [[ X ]] . But this holds only if A i contains a root of X j − j which implies A i is trivial as will shortly show after the following remark.Consider an algebra R over Q . Assume R contains a root of X n − n . Then letting Q [ x ] = Q [ X ] / (cid:104) X n − (cid:105) , one will have a homomorphism ξ : Q [ x ] → R . By Eisenstein’s criterion thepolynomial X n − Q , making Q [ x ] a field of dimension 2 n and ξ either an injectionwith a trivial kernel or ξ = Q [ x ] → i ∈ I , the algebra A i containsa root of X j − j . For each i ∈ I , let A i be of dimension m i over Q . We have that A i containsa root of X mi − Q ( mi √ ) → A i which since A i has dimension m i < m i means that A i is trivial for all i ∈ I . Hence, A i (cid:13) ⊥ and consequently C (cid:13) ⊥ . We have shown that for anyobject D of RA op Q if D (cid:13) ADC then D (cid:13) ⊥ . Hence Sh ( RA op Q , J ) (cid:13) ¬ ADC.As a consequence we get that the internal axiom of choice does not hold in Sh ( RA op Q , J ) . Let K be a field of characteristic 0. We consider a typed language L [ N , F ] K of the form described inSection 2 with two basic types N and F and the elements of the field K as its set of constants. Consider atheory T in the language L [ N , F ] K , such that T has as an axiom every atomic formula or the negation ofone valid in the field K , T equips N with the (Peano) axioms of natural numbers and equips F with theaxioms of a field containing K . If we interpret the types N and F by the objects (cid:101) N and F , respectively, inthe topos Sh ( RA opK , J ) then we have, by the results proved earlier, a model of T in Sh ( RA opK , J ) . LetAlgCl be the axiom schema of algebraic closure with quantification over the type F , then one has that T + AlgCl has a model in Sh ( RA opK , J ) with the same interpretation. Let φ be a sentence in the languagesuch that T + AlgCl (cid:96) φ in IHOL deduction system. By soundness [1] one has that Sh ( RA opK , J ) (cid:13) φ ,i.e. for all finite dimensional regular algebras R over K , R (cid:13) φ which is then a constructive interpretationof the existence of the algebraic closure of K .This model can be implemented, e.g. in Haskell. In the paper [12] by the authors, an algorithm forcomputing the Puiseux expansions of an algebraic curve based on this model is given. The statementwith the assumption of algebraic closure is: “ Let K be a field of characteristic and G ( X , Y ) = Y n + ∑ ni = b i ( X ) Y n − i ∈ K [[ X ]][ Y ] a monic, non-constant polynomial separable over K (( X )) . Let F be the algebraic closure of K, we have a positiveinteger m and a factorization G ( T m , Y ) = ∏ ni = ( Y − α i ) with α i ∈ F [[ T ]] ” We can then extract the following computational content “ Let K be a field of characteristic and G ( X , Y ) = Y n + ∑ ni = b i ( X ) Y n − i ∈ K [[ X ]][ Y ] a monic, non-constant polynomial separable over K (( X )) . Then there exist a (von Neumann) regular algebra R overK and a positive integer m such that G ( T m , Y ) = ∏ ni = ( Y − α i ) with α i ∈ R [[ T ]] ” For example applying the algorithm to G ( X , Y ) = Y − Y + XY + X ∈ Q [ X , Y ] we get a regular0 A Sheaf Model of the Algebraic Closure algebra Q [ b , c ] with b − / = c − = G ( X , Y ) =( Y + ( − b − ) X + ( − b − ) X + ( − b − ) X + ... )( Y + ( b − ) X + ( b − ) X + ( b − ) X + ... )( Y − c + X + cX + X + cX + X + ... )( Y + c + X − cX + X − cX + X + ... ) Another example of a possible application of this model is as follows: suppose one want to show that “For discrete field K, if f ∈ K [ X , Y ] is smooth, i.e. ∈ (cid:104) f , f x , f Y (cid:105) , then K [ X , Y ] / (cid:104) f (cid:105) is a Pr¨ufer ring.“ To prove that a ring is Pr¨ufer one needs to prove that it is arithmetical, that is ∀ x , y ∃ u , v , w [ yu = vx ∧ yw = ( − u ) x ] . Proving that K [ X , Y ] / (cid:104) f (cid:105) is arithmetical is easier in the case where K is algebraicallyclosed [3]. Let F be the algebraic closure of K in Sh ( RA opK , J ) . Now F [ X , Y ] / (cid:104) f (cid:105) being arithmeticalamounts to having a solution u , v , and w to a linear system yu = vx , yw = ( − u ) x . Having obtainedsuch solution, by Rouch´e–Capelli–Fonten´e theorem we can conclude that the system have a solution in K [ X , Y ] / (cid:104) f (cid:105) . Sh ( RA opK , J ) In this section we will demonstrate that in a classical metatheory one can show that the topos Sh ( RA opK , J ) is boolean. In fact we will show that, in a classical metatheory, the boolean algebra structure of the sub-object classifier is the one specified by the boolean algebra of idempotents of the algebras in RA K .Except for Theorem 8.8 the reasoning in this section is classical. Recall that the idempotents of a com-mutative ring form a boolean algebra with the meaning of the logical operators given by : (cid:62) = ⊥ = e ∧ e = e e , e ∨ e = e + e − e e and ¬ e = − e . We write e ≤ e iff e ∧ e = e and e ∨ e = e A sieve S on an object C is a set of morphisms with codomain C such that if g ∈ S and cod( h ) = dom( g )then gh ∈ S . A cosieve is defined dually to a sieve. A sieve S is said to cover a morphism f : D → C if f ∗ ( S ) = { g | cod( g ) = D , f g ∈ S } contains a cover of D . Dually, a cosieve M on C is said to cover amorphism g : C → D if the sieve dual to M covers the morphism dual to g . Definition 8.1 (Closed cosieve) . A sieve M on an object C of C is closed if for all f with cod( f ) = C if M covers f then f ∈ M . A closed cosieve on an object C of C op is the dual of a closed sieve in C . Fact 8.2 (Subobject classifier) . The subobject classifier in the category of sheaves on a site ( C , J ) isthe presheaf Ω where for an object C of C the set Ω ( C ) is the set of closed sieves on C and for eachf : D → C we have a restriction map M (cid:55)→ { h | cod( h ) = D , f h ∈ M } . Lemma 8.3.
Let R be an object of RA K . If R is a field the closed cosieves on R are the maximal cosieve { f | dom( f ) = R } and the minimal cosieve { R → } .Proof. Let S be a closed cosieve on R and let ϕ : R → A ∈ S and let I be a maximal ideal of A . If A is nontrivial we have a field morphism R → A / I in S where A / I is a finite field extension of R . Let A / I = R [ a , ..., a n ] . But then the morphism ϑ : R → R [ a , ..., a n − ] is covered by S . Thus ϑ ∈ S since S is closed. By induction on n we get that a field automorphism η : R → R is in S but then by compositionof η with its inverse we get that 1 R ∈ S . Consequently, any morphism with domain R is in S . Corollary 8.4.
For an object R of RA K . If R is a field, then Ω ( R ) is a 2-valued boolean algebra. . Mannaa & T. Coquand Proof.
This is a direct Corollary of Lemma 8.3. The maximal cosieve ( R ) correspond to the idempotent1 of R , that is the idempotent e such that, ker 1 R = (cid:104) − e (cid:105) . Similarly the cosieve { R → } correspond tothe idempotent 0. Corollary 8.5.
For an object A of RA K , Ω ( A ) is isomorphic to the set of idempotents of A and theHeyting algebra structure of Ω ( A ) is the boolean algebra of idempotents of A.Proof. Classically a finite dimension regular algebra over K is isomorphic to a product of field extensionsof K . Let A be an object of RA K , then A ∼ = F × ... × F n where F i is a finite field extension of K . Theset of idempotents of A is { ( d , ..., d n ) | ≤ j ≤ n , d j ∈ F j , d j = d j = } . But this is exactly the set Ω ( F ) × ... × Ω ( F n ) ∼ = Ω ( A ) . It is obvious that since Ω ( A ) is isomorphic to a product of boolean algebras,it is a boolean algebra with the operators defined pointwise. Theorem 8.6.
The topos Sh ( RA opK , J ) is boolean.Proof. The subobject classifier of Sh ( RA opK , J ) is 1 true −−→ Ω where for an object A of RA K one hastrue A ( ∗ ) = ∈ A .It is not possible to show that the topos Sh ( RA opK , J ) is boolean in an intuitionistic metatheory as weshall demonstrate. First we recall the definition of the Limited principle of omniscience (LPO for short).
Definition 8.7 (LPO) . For any binary sequence α the statement ∀ n [ α ( n ) = ] ∨ ∃ n [ α ( n ) = ] holds.LPO cannot be shown to hold intuitionistically. One can, nevertheless, show that it is weaker thanthe law of excluded middle [2]. Theorem 8.8.
Intuitionistically, if Sh ( RA opK , J ) is boolean then LPO holds.Proof.
Let α ∈ K [[ X ]] be a binary sequence. By Lemma 5.7 one has an isomorphism Λ : F [[ X ]] ∼ −→ F (cid:101) N .Let Λ K ( α ) = β ∈ F (cid:101) N ( K ) . Assume the topos Sh ( RA opK , J ) is boolean. Then one has K (cid:13) ∀ n [ β ( n ) = ] ∨ ∃ n [ β ( n ) = ] . By ∨ this holds only if there exist a cocover of K { ϑ i : K → A i | i ∈ I } ∪ { ξ j : K → B j | j ∈ J } such that B j (cid:13) ∀ n [( β ξ j )( n ) = ] for all j ∈ J and A i (cid:13) ∃ n [( β ϑ i )( n ) = ] for all i ∈ I . Note that at leastone of I or J is nonempty since K is not covered by the empty cover.For each i ∈ I there exist a cocover { η (cid:96) : A i → D (cid:96) | (cid:96) ∈ L } of A i such that for all (cid:96) ∈ L , we have D (cid:96) (cid:13) ( β ϑ i η (cid:96) )( m ) = m ∈ (cid:101) N ( D (cid:96) ) . Let m = ∑ t ∈ T e t n t then we have a cocover { ξ t : D (cid:96) → C t = D (cid:96) / (cid:104) − e t (cid:105) | t ∈ T } such that C t (cid:13) ( β ϑ i η (cid:96) ξ t )( n t ) = ξ t η (cid:96) ϑ i ( α ( n t )) =
1. For each t wecan check whether α ( n t ) =
1. If α ( n t ) = ∃ n [ α ( n ) = ] . Otherwise, we have α ( n t ) = ξ t η (cid:96) ϑ i ( ) =
1. Thus the map ξ t η (cid:96) ϑ i : K → C t from the field K cannot be injective, whichleaves us with the conclusion that C t is trivial. If for all t ∈ T , C t is trivial then D (cid:96) is trivial as well.Similarly, if for every (cid:96) ∈ L , D (cid:96) is trivial then A i is trivial as well. At this point one either have either(i) a natural number m such that α ( m ) = ∃ n [ α ( n ) = ] . Or (ii) wehave shown that for all i ∈ I , A i is trivial in which case we have a cocover { ξ j : K → B j | j ∈ J } suchthat B j (cid:13) ∀ n [( β ξ j )( n ) = ] for all j ∈ J . Which by LC means K (cid:13) ∀ n [ β ( n ) = ] which by ∀ meansthat for all arrows K → R and elements d ∈ (cid:101) N ( R ) , R (cid:13) β ( d ) =
0. In particular for the arrow K K −→ K and every natural number m one has K (cid:13) β ( m ) = K (cid:13) α ( m ) =
0. By = we get that ∀ m ∈ N [ α ( m ) = ] . Thus we have shown that LPO holds. Corollary 8.9.
It cannot be shown in an intuitionistic metatheory that the topos Sh ( RA opK , J ) is boolean. A Sheaf Model of the Algebraic Closure
References [1] Steven Awodey (1997):
Logic in topoi: Functorial Semantics for Higher-Order Logic . Ph.D. thesis, TheUniversity of Chicago.[2] Douglas Bridges & Fred Richman (1987):
Varieties of Constructive Mathematics . Lecture note series, Cam-bridge University Press, doi:10.1017/cbo9780511565663.[3] Thierry Coquand, Henri Lombardi & Claude Quitt´e (2010):
Curves and coherent Pr¨ufer rings . J. Symb.Comput.
Dynamical method in algebra: effective Null-stellens¨atze . Annals of Pure and Applied Logic
About a new method for computing inalgebraic number fields . In Bob Caviness, editor:
EUROCAL ’85 , Lecture Notes in Computer Science
Effective Procedures in Field Theory . Philosophical Transactionsof the Royal Society of London. Series A, Mathematical and Physical Sciences
Sketches of an Elephant: A Topos Theory Compendium - Volume 2 . Oxford LogicGuides
44, Oxford University Press.[8] John F. Kennison (1982):
Separable algebraic closure in a topos . Journal of Pure and Applied Algebra
Alg`ebre Commutative, M´ethodes Constructives . Math´ematiquesen devenir, Calvage et Mounet.[10] Saunders MacLane & Ieke Moerdijk (1992):
Sheaves in Geometry and Logic: A First Introduction to ToposTheory , corrected edition. Springer, doi:10.1007/978-1-4612-0927-0.[11] Michael Makkai & Gonzalo E. Reyes (1977):
First order categorical logic: model-theoretical meth-ods in the theory of topoi and related categories . Lecture notes in mathematics
Dynamic Newton-Puiseux theorem . J. Logic & Analysis
An intuitionistic theory of types . Reprinted in Twenty-five years of constructive typetheory, Oxford University Press, 1998, 127–172.[14] Ray Mines, Fred Richman & Wim Ruitenburg (1988):
A course in constructive algebra . Universitext (1979),Springer-Verlag, doi:10.1007/978-1-4419-8640-5.[15] Andrej ˇSˇcedrov (1984):