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 possible constructive interpretation of the existence of the algebraic closure of a field in characteristic 0 by building, 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 based on this model.

gh j factors through f` for some ` 2 I. A family S 2 J(C) is called elementary cover or elementary covering family of C. A site is a category with coverage (C ; J). For a presheaf P : C op ! Set and family S = fgi : Ai ! Agi2I of morphisms of C we say that a family fsi 2 P(Ai)gi2I is compatible if for each `; j 2 I whenever we have h : B ! A` and f : B ! A j such that g`h = g j f then s`h = s j f , where by s`h we mean the restriction of s` along h, i.e. P(h)s`. A presheaf P is a sheaf on the site (C ; J) if for any object C and any f fi : Ci ! Cgi2I 2 J(C) if fsi 2 P(Ci)gi2I is compatible then there exist a unique s 2 P(C) such that s fi = si. We call such s the amalgamation of fsigi2I. Let J be a coverage on C we define a closure J of J as follows: For all objects C of C i. fC 1!C Cg 2 J (C), ii. If S 2 J(C) then S 2 J (C), and, iii. If fCi !fi Cgi2I 2 J (C) and for each i 2 I, fCi j g!ij Cig j2Ji 2 J (Ci) then fCi j fig!ij Cgi2I; j2Ji 2 J (C). A family T 2 J (C) is called cover or covering family of C.

We work with a typed language with equality L [V1; :::;Vn] having the basic types V1; :::;Vn and type
formers ; ( ) ; P( ). The language L [V1; :::;Vn] has typed constants and function symbols. For
any type Y one has a stock of variables y1; y2; ::: of type Y . Terms and formulas of the language are
defined as usual. We work within the proof theory of intuitionistic higher-order logic (IHOL). A detailed
description of this deduction system is given in [

The language L [V1; :::;Vn] along with deduction system IHOL can be interpreted in an elementary
topos in what is referred to as topos semantics. For a sheaf topos this interpretation takes a simpler form
reminiscent of Beth semantics, usually referred to as Kripke?Joyal sheaf semantics. We describe this
semantics here briefly following [

Let E = Sh(C ; J) be a sheaf topos. An interpretation of the language L [V1; :::;Vn] in the topos E is given as follows: Associate to each basic type Vi of L [V1; :::;Vn] an object Vi of E . If Y and Z are types of L [V1; :::;Vn] interpreted by objects Y and Z, respectively, then the types Y Z;Y Z; P(Z) are interpreted by Y Z; YZ; WZ, respectively, where W is the subobject classifier of E . A constant e of type E is interpreted by an arrow 1 !e E where E is the interpretation of E. For a term t and an object X of E , we write t : X to mean t has a type X interpreted by the object X.

Let f (x1; :::; xn) be a formula with variables x1 :X1; :::; xn :Xn. Let c1 2 X j(C); :::; cn 2 Xn(C) for some object C of C . We define the relation C forces f (x1; :::; xn)[c1; :::; cn] written C f (x1; :::; xn)[c1; :::; cn] by induction on the structure of f .

Definition 2.1 (Forcing). First we replace the constants in f by variables of the same type as follows: Let e1 : E1; :::; em : Em be the constants in f (x1; :::; xn) then C f (x1; :::; xn)[c1; :::; cn] iff C

f [y1=e1; :::; ym=em](y1; :::; ym; x1; :::; xn)[e1C ( ); :::; emC ( ); c1; :::; cn] where yi : Ei and ei : 1 ! Ei is the interpretation of ei.

Now it suffices to define the forcing relation for formulas free of constants by induction as follows: > C ? C = C ^ C _ C

Ci ) C

D >. ? iff the empty family is a cover of C. (x1 = x2)[c1; c2] iff c1 = c2. (f ^ y)(x1; :::; xn)[c1; :::; cn] iff C f (x1; :::; xn)[c1; :::; cn] and C y(x1; :::; xn)[c1; :::; cn]. (f _ y)(x1; :::; xn)[c1; :::; cn] iff there exist a cover fCi !fi Cgi2I 2 J (C) such that f (x1; :::; xn)[c1 fi; :::; cn fi] or Ci y(x1; :::; xn)[c1 fi; :::; cn fi] for each i 2 I. (f ) y)(x1; :::; xn)[c1; :::; cn] iff for every morphism f : D ! C whenever f (x1; :::; xn)[c1 f ; :::; cn f ] one has D y(x1; :::; xn)[c1 f ; :::; cn f ]. Let y be a variable of the type Y interpreted by the object Y of E .

9 C (9yf (x1; :::; xn; y))[c1; :::; cn] iff there exist a cover fCi !fi Cgi2I 2 J (C) such that for each i 2 I one has Ci f (x1; :::; xn; y)[c1 fi; :::; cn fi; d] for some d 2 Y(Ci). (8yf (x1; :::; xn; y))[c1; :::; cn] iff for every morphism f : D ! C and for all d 2 Y(D) one has f (x1; :::; xn; y)[c1 f ; :::; cn f ; d].

We have the following derivable local character and monotonicity laws: LC If fCi !fi Cgi2I 2 J (C) and for all i 2 I, Ci f (x1; :::; xn)[c1 fi; :::; cn fi] then C f (x1; :::; xn)[c1; :::; cn]. f (x1; :::; xn)[c1; :::; cn] and f : D ! C then D f (x1; :::; xn)[c1 f ; :::; cn f ].

Definition 3.1 (Regular ring). A commutative ring R is (von Neumann) regular if for every element a 2 R there exist b 2 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 refer to the quasi-inverse of a. A ring is regular iff it is zero-dimensional and reduced. To be regular is equivalent to the fact that any principal ideal (consequently, any finitely generated ideal) is generated by an idempotent. If R is regular and a 2 R then e = aa is an idempotent such that hei = hai and R is isomorphic to R0 R1 with R0 = R=hei and R1 = R=h1 ei. Furthermore a is 0 on the component R0 and invertible on the component R1.

Definition 3.2 (Fundamental system of orthogonal idempotents). A family (ei)i2I of idempotents in a ring R is a fundamental system of orthogonal idempotents if ċi2I ei = 1 and 8i; j[i 6= j ) eie j = 0].

composition A = Ġi2I A=h1 eii.

Proof. Follows by induction from the fact that A = A=hei A=h1 ei for an idempotent e 2 A.

Definition 3.4 (Separable polynomial). Let R be a ring. A polynomial p 2 R[X ] is separable if there exist r; s 2 R[X ] such that r p + sp0 = 1, where p0 2 R[X ] is the derivative of p.

Definition 3.5. A ring R is a (strict) Be´zout ring if for all a; b 2 R we can find g; a1; b1; c; d 2 R such that a = a1g, b = b1g and ca1 + db1 = 1 [9, Ch. 4].

If R is a regular ring then R[X ] is a strict Be´zout ring (and the converse is true [

R/hai

R[ 1a ] (ii.)

R

, for a 2 R. Dualized, these are elementary covering families of R in C op. We

An algebra A over a field K is finitely presented if it is of the form K[X1; ::; Xn]=h f1; :::; fmi, i.e. the quotient 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 consider the category of all finitely presented T0 algebras where T0 is an equational subtheory of T . The axioms of T then give rise to a coverage on the dual category [11, Ch. 9]. For our purpose consider the category C of finitely presented K-algebras. Given an object R of C , the axiom schema of algebraic closure and the field axiom give rise to families (i.) R ! R[X ]=hpi where p 2 R[X ] is monic and observe however that we can limit our consideration only to those finitely presented K-algebras that are zero dimensional and reduced, i.e. regular. In this case we can assume a is an idempotent and we only consider extensions R[X ]=hpi where p is separable.

Let RA K be the small category of finitely presented regular algebras over a fixed field K and Khomomorphisms. First we fix an countable set of names S. An object of RA K is a regular algebra of the form K[X1; :::; Xn]=h f1; :::; fmi where Xi 2 S for all 1 i n. Note that for any object R, there is a unique morphism K ! R. A finitely presented regular K-algebra A is a finite dimensional K-algebra, i.e. A has a finite dimension as a vector space over K [9, Ch 4, Theorem 8.16]. The trivial ring 0 is the terminal object in the category RA K and K is its initial object.

To specify a coverage J on the category RA oKp, we define for each object A a collection Jop(A) of families of morphisms of RA K with domain A. We then take J(A) to be the dual of Jop(A) in the sense that fji : Ai ! Agi2I 2 J(A) if and only if fji : A ! Aigi2I 2 Jop(A) where ji of RA K is the dual of ji of RA oKp. We call Jop cocoverage. We call an element of Jop(A) an elementary cocover (cocovering family) 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 ]=hpi where p 2 R[X ] is non-constant, monic and separable.

Definition 3.7 (Topology for RA oKp). For an object A of RA K the cocovering families are given by: (i.) If (ei)i2I is a fundamental system of orthogonal idempotents of A, then fA j!i A=h1 eiigi2I 2

Jop(A) where for each i 2 I, ji is the canonical homomorphism. (ii.) Let A[a] be a separable extension of A. We have fA !J A[a]g 2 Jop(A) where J is the canonical embedding.

Note that in particular 3.7.(i.) implies that the trivial algebra 0 is covered by the empty family of morphisms since an empty family of elements in this ring form a fundamental system of orthogonal idempotents. Also note that 3.7.(ii.) implies that fA 1!A Ag 2 Jop(A).

Lemma 3.8. The function J of Definition 3.7 is a coverage on RA oKp.

Proof. Let h : R ! A be a morphism of RA K and S 2 Jop(R). We show that there exist an elementary cocover T 2 Jop(A) such that for each J 2 T , J h factors through some j 2 S. By duality, this implies J is a coverage on RA oKp. By case analysis on the clauses of Definition 3.7. (i.) If S = fji : R ! R=h1 eiigi2I, where (ei)i2I is a fundamental system of orthogonal idempotents of R. In A, the family (h(ei))i2I is fundamental system of orthogonal idempotents. We have an elementary cocover fJi : A ! A=h1 h(ei)igi2I 2 Jop(A). For each i 2 I, the homomorphism h induces a K-homomorphism hei : R=h1 eii ! A=h1 h(ei)i where hei (r + h1 eii) = h(r) + h1 h(ei)i. Since Ji(h(r)) = h(r) + h1 h(ei)i we have that Jih = hei ji. (ii.) If S = fj : R ! R[r]g with R[r] = R[X ]=hpi and p 2 R[X ] monic, non-constant, and separable.

Since sp +t p0 = 1, we have h(s)h(p) + h(t)h(p0) = h(s)h(p) + h(t)h(p)0 = 1. Then q = h(p) 2 A[X ] is separable. Let A[a] = A[X ]=hqi. We have an elementary cocover fJ : A ! A[a]g 2 Jop(A) where J is the canonical embedding. Let z : R[r] ! A[a] be the K-homomorphism such that z jR = h and z (r) = a. For b 2 R, we have J (h(b)) = z (j(b)).

Lemma 3.9. Let P : RA K ! Set be a presheaf on RA oKp such that P(0) = 1. Let R be an object of RA K and let (ei)i2I be a fundamental system of orthogonal idempotents of R. For each i 2 I, let Ri = R=h1 eii and let ji : R ! Ri be the canonical homomorphism. Any family fsi 2 P(Ri)g is compatible. Proof. Let B be an object and for some i; j 2 I let J : Ri ! B and z : R j ! B be such that J ji = z j j. We will show that P(J )(si) = P(z )(s j).

(i.) If i = j, then since ji is surjective we have J = z and P(J ) = P(z ). (ii.) If i 6= j, then since eie j = 0, ji(ei) = 1 and j j(e j) = 1 we have j j(ei) = j j(eie j) = 0. But then 1 = J (1) = J (ji(ei)) = z (j j(ei)) = z (0) = 0

Hence B is the trivial algebra 0. By assumption P(0) = 1, hence P(J )(si) = P(z )(s j) = . Corollary 3.10. Let F be a sheaf on (RA oKp; J). Let R be an object of RA K and (ei)i2I a fundamental system of orthogonal idempotents of R. Let Ri = R=h1 eii and ji : R ! Ri be the canonical homomorphism. The map f : F(R) ! Ġi2I F(Ri) such that f (s) = (F(ji)s)i2I is an isomorphism. Proof. Since F(0) = 1, by Lemma 3.9 any family fsi 2 F(Ri)gi2I is compatible. Since F is a sheaf, the family fsi 2 F(Ri)gi2I has a unique amalgamation s 2 F(R) with restrictions sji = si. The isomorphism is given by f s = (sji)i2I. We can then use the tuple notation (si)i2I to denote the element s in F(R).

One say that a polynomial f 2 R[X ] has a formal degree n if f can be written as f = anX n + ::: + a0 which is to express that for any m > n the coefficient of X m is known to be 0.

Proof. The statement follows immediately since the R-basis ai; i > 0 and b j; j > 0 are linearly independent.

Corollary 3.12. Let R be an object of RA K and p 2 R[X ] separable and monic. Let R[a] = R[X ]=hpi and j : R ! R[a] the canonical morphism. Let R[b; c] = R[X ;Y ]=hp(X ); p(Y )i. The commuting diagram ?

R[b, c] ?

R[a] is a pushout diagram of RA K. Moreover, j is the equalizer of z and J .

B be morphisms of RA K such that hj = rj. Then for all r 2 R we have h(r) =

A be a pushout diagram. A family fs 2 F(R[a])g is compatible if and only if sJ = sh. Corollary 3.13. The coverage J is subcanonical, i.e. all representable presheaves in SetRA K are sheaves on (RA oKp; J). 4

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 j : A ! C of RA K, F(j) = j.

Lemma 4.1. F is a sheaf of sets on the site (RA oKp; J) Proof. By case analysis on the clauses of Definition 3.7.

is a compatible family. By the isomorphism R (ji)i!2I Ġi2I R=h1 is the unique element such that ji(a) = ai. (i.) Let fR !ji R=h1 eiigi2I 2 Jop(R), where (ei)i2I is fundamental system of orthogonal idempotents of R. The presheaf F has the property F(0) = 1. By Lemma 3.9 a family fai 2 R=h1 eiigi2I eii the element a = (ai)i2I 2 R (ii.) Let fR !j R[a]g 2 Jop(R) where R[a] = R[X ]=hpi with p 2 R[X ] monic, non-constant and ? ? separable polynomial. Let fr 2 R[a]g be a compatible family. Let R ? the pushout diagram of Corollary 3.12. Compatibility then implies J (r) = z (r) which by the same Corollary is true only if the element r is in R. We then have that r is the unique element restricting to itself along the embedding j.

R[b, c] be R[a]

We fix a field K of characteristic 0. Let L [F; +; :] be a language with basic type F and function symbols +; : : F F ! F. We extend L [F; +; :] by adding a constant symbol of type F for each element a 2 K, to obtain L [F; +; :]K. Define Diag(K) as : if f is an atomic L [F; +; :]K-formula or the negation of one such that K j= f (a1; :::; an) then f (a1; :::; an) 2 Diag(K). The theory T equips the type F with axioms 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) 8x [0 + x = x + 0 = x] (b) 8x8y8z[x + (y + z) = (x + y) + z] (c) 8x9y[x + y = 0] (d) 8x8y[x + y = y + x] 3. The axioms of a commutative ring: (a) 8x [x1 = x] (b) 8x [x0 = 0] (c) 8x8y[xy = yx] (d) 8x8y8z[x(yz) = (xy)z] (e) 8x8y8z[x(y + z) = xy + xz] 4. The field axioms: (a) 1 6= 0. (b) 8x[x = 0 _ 9y[xy = 1]]. 5. The axiom schema for algebraic closure: 8a1 : : : 8an9x[xn + ċin=1 xn iai = 0]. 6. F is algebraic over K: 8x[Wp2K[Y ] p(x) = 0].

With these axioms the type F becomes the type of an algebraically closed field containing K. We proceed to show that with the interpretation of the type F by the object F the topos Sh(RA oKp; J) is a model 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 2 K gives rise to a unique map 1 !a F, that is the map 7! a 2 F(K). Every constant a 2 K of the language is then interpreted by the corresponding unique arrow 1 !a F. (we use the same symbol for constants and their interpretation to avoid cumbersome notation). That F satisfies Diag(K) then follows directly. Lemma 4.3. F is a ring object.

Lemma 4.4. F is a field.

Proof. For an object C of RA K the object F(C) is a commutative ring.

Proof. For any object R of RA K one has R 1 6= 0 since for any R !j C such that C 1 = 0 one has that C is trivial and thus C ?. Next we show that for variables x and y of type F and any object R of RA oKp we have R 8x [x = 0 _ 9y [xy = 1]]. Let j : A ! R be a morphism of RA oKp and let a 2 A. We need to show that A a = 0 _ 9y[ya = 1]. The element e = aa is an idempotent and we have a cover fj1 : A=hei ! A; j2 : A=h1 ei ! Ag 2 J (A) with A=hei aj1 = 0 and A=h1 ei (aj2)(a j2) = ej2 = 1. Hence by 9 we have A=h1 ei 9y[(aj2)y = 1] and by _ , A=h1 ei aj2 = 0 _ 9y[(aj2)y = 1]. Similarly, A=hei aj1 = 0 _ 9y[(aj1)y = 1]. By 8 we get R 8x [x = 0 _ 9y [xy = 1]].

To show that A 8a1 : : : 8an9x [xn + ċin=1 xn iai = 0] for every n, we need to be able to extend an algebra R of RA K with the appropriate roots. We need the following lemma.

Lemma 4.5. Let L be a field and f 2 L[X ] a monic polynomial. Let g = h f ; f 0i, where f 0 is the derivative of 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 h0. We have h = l1a. Let d be the gcd of a and a0. We have a = l2d and a0 = m2d, with l2 and m2 coprime.

The polynomial a divides h0 = l1a0 + l10a and hence that a = l2d divides l1a0 = l1m2d. It follows that l2 divides l1m2 and since l2 and m2 are coprime, that l2 divides l1.

Also, if an divides p then p = qan and p0 = q0an + nqa0an 1. Hence dan 1 divides p0. Since l2 divides l1, this implies that an = l2dan 1 divides l1 p0. So an+1 divides al1 p0 = hp0.

Since a divides f and f 0, a divides g. We show that an divides g for all n by induction on n. If an divides g we have just seen that an+1 divides g0h. Also an+1 divides h0g since a divides h0. So an+1 divides g0h + h0g = f 0. On the other hand, an+1 divides f = hg = l1ag. So an+1 divides g which is the gcd of f and f 0. 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 oKp we have R 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 0 its derivative. There is a cocover fji : R ! Rigi2I 2 J op(R) and for each Ri we have h; g; q; r; s 2 Ri[X ] such that ji( f ) = hg, ji( f 0) = qg and rh + sq = 1. Moreover, h is monic and separable.

Note that in characteristic 0, if f is monic and non-constant the separable associate of f is nonconstant.

Lemma 4.7. The field object F 2 Sh(RA oKp; J) is algebraically closed.

Proof. We prove that for all n > 0 and all (a1; :::; an) 2 Fn(R) = Rn, one has R 9x [xn + ċin=1 xn iai = 0]. Let f = xn + ċin=1 xn iai. By Corollary 4.6 we have a cover fJ j : R j ! Rg j2I 2 J (R) such that in each R j we have g = h f J j; f 0J ji and f J j = hg with h 2 R j[X ] monic and separable. Note that if deg f 1, h is non-constant. For each R j we have a singleton cover fj : R j[b] ! R j j R j[b] = R j[X ]=hhig 2 J (R j). That is, we have R j[b] bn + ċin=1 bn 1(aiJ jj) = 0. By 9 we get R j[b] 9x [xn + ċin=1 xn 1(aiJ jj) = 0] and by LC we have R j 9x [xn + ċin=1 xn 1(aiJ j) = 0]. Since this is true for each R j; j 2 J we have by LC R 9x [xn + ċin=1 xn 1ai = 0].

Lemma 4.8. F is algebraic over K.

Proof. We will show that for any object R of RA K and element r 2 R one has R Wp2K[X] p(r) = 0. Since R is a finitely presented K-algebra we have that R is a finite integral extension of a polynomial ring K[Y1; :::;Yn] R where Y1; ::;Yn are elements of R algebraically independent over K and that R has Krull dimension n [9, Ch 13, Theorem 5.4]. Since R is zero-dimensional (i.e. has Krull dimension 0) we have n = 0 and R is integral over K, i.e. any element r 2 R is the zero of some monic polynomial over K. 5

Here we describe the object of natural numbers in the topos Sh(RA oKp; J) and the object of power series over the field F. This will be used in section 6 to show that the axiom of dependent choice does not hold when 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. That j is, P(A) = B and P(A ! R) = 1B for all objects A and all morphism j of RA K. Let Pe : RA K ! Set be the presheaf such that Pe(A) is the set of elements of the form f(ei; bi)gi2I where (ei)i2I is a fundamental system of orthogonal idempotents of A and for each i, bi 2 B. We express such an element as a formal sum ċi2I eibi. Let j : A ! R be a morphism of RA K, the restriction of ċi2I eibi 2 Pe(A) along j is given by (ċi2I eibi)j = ċi2I j(ei)bi 2 Pe(R). In particular with canonical morphisms ji : A ! A=h1 eii, one has for any j 2 I that (ċi2I eibi)j j = b j 2 Pe(A=h1 e ji). Two elements ċi2I eibi 2 Pe(A) and ċ j2J d jc j 2 Pe(A) are equal if and only if 8i 2 I; j 2 J[bi 6= c j ) eid j = 0].

To prove that Pe is a sheaf we will need the following lemmas.

Proof. In R one has ċ j2Ji eid j = ei ċ j2Ji d j = ei(1 + h1 eii) = ei. Hence, ċ eid j = ċ ei = 1. For i2I; j2Ji i2I some i 2 I and t; k 2 Ji we have (eidt )(eidk) = ei(0 + h1 eii) = 0 in R. Thus for i; ` 2 I, j 2 Ji and s 2 J` one has i 6= ` _ j 6= s ) (eid j)(e`ds) = 0.

polynomial of degree m > n. If in R[X ;Y ] one has f (Y )(1 f (X )) = 0 mod hp(X ); p(Y )i then f = e 2 R with e an idempotent.

Proof. Let f (Z) = ċin=0 riZi. By the assumption, for some q; g 2 R[X ;Y ] f (Y )(1

n f (X )) = ċ ri(1 i=0 n ċ r jX j)Y i = qp(X ) + gp(Y ) j=0 One has ċin=0 ri(1 ċnj=0 r jX j)Y i = g(X ;Y )p(Y ) mod hp(X )i. Since p(Y ) is monic of Y -degree greater than n, one has that ri(1 ċnj=0 r jX j) = 0 mod hp(X )i for all 0 i n. But this means that rirnX n + rirn 1X n 1 + ::: + rir0 ri 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 ri2 = 0 and hence ri = 0 since R is reduced. For i = 0 we have r0r0 r0 = 0 and thus r0 is an idempotent of R. Lemma 5.3. The presheaf Pe described above is a sheaf on (RA oKp; J).

Proof. By case analysis on Definition 3.7.

(i.) Let fR !ji R=h1 eiigi2I 2 Jop(R) where (ei)i2I be a fundamental system of orthogonal idempotents of an object R. Let R=h1 eii = Ri. Since Pe(0) = 1 by Lemma 3.9 any set fsi 2 Pe(Ri)gi2I is compatible. For each i, Let si = ċ j2Ji [d j]b j. By Lemma 5.1 we have an element s = ċ (eid j)b j 2 Pe(R) the restriction of which along ji is the element ċ j2Ji [d j]b j 2 Pe(Ri).

i2I; j2Ji It remains to show that this is the only such element. Let there be an element ċ`2L c`a` 2 Pe(R) that restricts to ui = si along ji. We have ui = ċ`2L[c`]a`. One has that for any j 2 Ji and ` 2 L, b j 6= a` ) [c`d j] = 0 in Ri, hence, in R one has b j 6= a` ) c`d j = r(1 ei). Multiplying both sides of c`d j = r(1 ei) by ei we get b j 6= a` ) c`(eid j) = 0. Thus proving s = ċ`2L c`a`. (ii.) Let fj : R ! R[a] = R[X ]=hpig 2 Jop(R) where p 2 R[X ] is monic non-constant and separable. Let the singleton fs = ċi2I eibi 2 Pe(R[a])g be compatible. We can assume w.l.o.g. that 8i; j 2 I [i 6= j ) bi 6= b j] since if bk = b` one has that (ek + e`)bl + ċ jj62=I`; j6=k e jb j = s. (Note that an idempotent ei of R[a] is a polynomial ei(a) in a of formal degree less than deg p). Let R[c; d] = ? R[X ;Y ]=hp(X ); p(Y )i, by Corollary 3.12, one has a pushout diagram R ? where z jR = J jR = 1R, z (a) = d and J (a) = c. That the singleton fsg is compatible then means sJ = ċi2I ei(c)bi = sz = ċi2I ei(d)bi, i.e. 8i; j 2 I [bi 6= b j ) ei(c)e j(d) = 0]. By the assumption that bi 6= b j whenever i 6= j we have in R[c; d] that e j(d)ei(c) = 0 for any i 6= j 2 I. Thus e j(d) ċi6= j ei(c) = e j(d)(1 e j(c)) = 0, i.e. in R[X ;Y ] one has e j(Y )(1 e j(X )) = 0 mod hp(X ); p(Y )i. By Lemma 5.2 we have that e j(X ) = e j(Y ) = e 2 R. We have thus shown s is R[c, d] ?

R[a] equal to ċ j2J d jb j 2 Pe(R[a]) such that d j 2 R for j 2 J. That is ċ j2J d jb j 2 Pe(R). Thus we have found a unique (since Pe(j) is injective) element in Pe(R) restricting to s along j.

GR(b) = b 2 Pe(R) for any object R and b 2 B. If E is a sheaf and L : P ! E is a morphism of presheaves, then there exist a unique sheaf morphism D : Pe ! E such that the following diagram, of SetRA K , comP

? P e ? ?

E mutes.

Proof. Let a = ċi2I eibi 2 Pe(A) and let Ai = A=h1 eii with canonical morphisms ji : A ! Ai.

Let E and L be as in the statement of the lemma. If there exist a sheaf morphism D : Pe ! E, then D being a natural transformation forces us to have for all i 2 I, E(ji)DA = DAi Pe(ji). By Lemma 3.10, we know that the map d 2 E(A) 7! (E(ji)d 2 E(Ai))i2I is an isomorphism. Thus it must be that DA(a) = (DAi Pe(ji)(a))i2I = (DAi (bi))i2I. But DAi (bi) = DAi GAi (bi). To have DG = L we must have DAi (bi) = LAi (bi). Hence, we are forced to have DA(a) = (LAi (bi))i2I. Note that D is unique since its value DA(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 SetRA K . We associate to N a sheaf Ne as described above. From Lemma 5.4 one can easily show that Ne satisfy the axioms of a natural numbers object in Sh(RA oKp; J).

Definition 5.5. Let F[[X ]] be the presheaf mapping each object R of RA K to F[[X ]](R) = R[[X ]] = RN 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 FNe .

Proof. Let C be an object of RA oKp. Since FNe(C) = yC Ne ! F, an element aC 2 FNe(C) is a family of elements of the form aC;D : yC(D) Ne(D) ! F(D) where D is an object of RA oKp. Define Q : FNe ! F[[X ]] as (Qa)C(n) = aC;C(1C; n). Define L : F[[X ]] ! FNe as (Lb )C;D(C !j D; ċ eini) = (Jij(bC(ni)))i2I 2 F(D)

i2I where D J!i D=h1 eii is the canonical morphism. Note that by Lemma 3.10 one indeed has that (Jij(bC(ni)))i2I 2 Ġi2I F(Di) = F(D). One can easily verify that Q and L are natural. It remains to show the isomorphism. One one hand we have (LQa)C;D(j; ċ eini) = (Jij((Qa)C(ni)))i2I = (Jij(aC;C(1C; ni)))i2I i2I = ((aC;Di (Jij; ni)))i2I = aC;D(j; ċ eini) i2I Thus showing LQ = 1FNe . On the other hand, (QLb )C(n) = (Lb )C;C(1C; n) = 1C1C(bC(n)) = bC(n). Thus QL = 1F[[X]]. Lemma 5.8. The power series object F[[X ]] is a ring object.

Proof. A Corollary to Lemma 4.3. 6

The (external) axiom of choice fails to hold (even in a classical metatheory) in the topos Sh(RA oKp; J) whenever the field K is not algebraically closed. To show this we will show that there is an epimorphism in Sh(RA oKp; J) with no section.

each object C of C and each element c 2 G(C) there is a cover S of C such that for all f : D ! C in the cover S the element c f is in the image of QD. [10, Ch. 3].

Lemma 6.2. Let K be a field of characteristic 0 not algebraically closed. There is an epimorphism in Sh(RA oKp; J) with no section.

Proof. Let f = X n + ċin=1 riX 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 L : F ! F defined by LC(c) = cn + ċin=11 ricn i 2 C. Given d 2 F(C), let g = X n + ċin=11 riX n i d. By Corollary 4.6 there is a cover fC` j!` Cg`2L 2 J (C) with h` 2 C`[X ] a separable non-constant polynomial dividing g. Let C`[x`] = C`[X ]=hh`i one has a singleton cover fC`[x`] J!` C`g and thus a composite cover fC`[x`] J`j!` Cg`2L 2 J (C). Since x` is a root of h` j g we have LC`[x`](x`) = x`n + ċin=11 rix`n i = d or more precisely LC`[x`](x`) = dj`J`. Thus, L is an epimorphism (by Fact 6.1) and it has no section, for if it had a section Y : F ! F then one would have n YK( rn) = a 2 K such that an + ċi=1 rian i = 0 which is not true by assumption.

Theorem 6.3. Let K be a field of characteristic 0 not algebraically closed. The axiom of choice fails to hold in the topos Sh(RA oKp; J).

We note that in Per Martin-Lo¨f type theory one can show that (see [

(Ġ x 2 A)(ċ y 2 B[x])C[x; y] ) (ċ f 2 (Ġ x 2 A)B[x])(Ġ x 2 A)C[x; f (x)] As demonstrated in the topos Sh(RA oKp; J) we have an example of an intuitionistically valid formula of the form 8x9yf (x; y) where no function f exist for which 9 f 8xf (x; f (x)) holds.

We demonstrate further that when the base field is Q the weaker axiom of dependent choice does not hold (internally) in the topos Sh(RA oQp; J). For a relation R Y Y the axiom of dependent choice is stated as 8x9yR(x; y) ) 8x9g 2 Y N [g(0) = x ^ 8nR(g(n); g(n + 1))] (ADC) Theorem 6.4. Sh(RA oQp; J)

:ADC.

Proof. Consider the binary relation on the algebraically closed object F defined by the characteristic function f (x; y) := y2 x = 0. Assume C ADC for some object C of RA K. Since C 8x9y[y2 x = 0] we have C 8x9g 2 FNe[g(0) = x ^ 8n[g(n)2 = g(n + 1)]]. That is for all morphisms C !z A of RA K and elements a 2 F(A) one has A 9g 2 FNe [g(0) = a ^ 8n[g(n)2 = g(n + 1)]]. Taking a = 2 we have A 9g 2 FNe [g(0) = 2 ^ 8n[g(n)2 = g(n + 1)]]. Which by 9 implies the existence of a cocover fhi : A ! Ai j i 2 Ig and power series ai 2 FNe (Ai) such that Ai ai(0) = 2 ^ 8n[ai(n)2 = ai(n + 1)]]. By Lemma 5.7 we have FNe (Ai) = Ai[[X ]] and thus the above forcing implies the existence of a series ai = 2 + 21=2 + ::: + 21=2j + ::: 2 Ai[[X ]]. But this holds only if Ai contains a root of X 2j 2 for all j which implies Ai is trivial as will shortly show after the following remark.

Consider an algebra R over Q. Assume R contains a root of X 2n 2 for some n. Then letting Q[x] = Q[X ]=hX 2n 2i, one will have a homomorphism x : Q[x] ! R. By Eisenstein?s criterion the polynomial X 2n 2 is irreducible over Q, making Q[x] a field of dimension 2n and x either an injection with a trivial kernel or x = Q[x] ! 0.

Now we continue with the proof. Until now we have shown that for all i 2 I, the algebra Ai contains a root of X 2j 2 for all j. For each i 2 I, let Ai be of dimension mi over Q. We have that Ai contains a root of X 2mi 2 and we have a homomorphism Q( 2mpi2) ! Ai which since Ai has dimension mi < 2mi means that Ai is trivial for all i 2 I. Hence, Ai ? and consequently C ?. We have shown that for any object D of RA oQp if D ADC then D ?. Hence Sh(RA oQp; J) :ADC.

As a consequence we get that the internal axiom of choice does not hold in Sh(RA oQp; J). 7

Let K be a field of characteristic 0. We consider a typed language L [N; F]K of the form described in
Section 2 with two basic types N and F and the elements of the field K as its set of constants. Consider a
theory T in the language L [N; F]K, such that T has as an axiom every atomic formula or the negation of
one valid in the field K, T equips N with the (Peano) axioms of natural numbers and equips F with the
axioms of a field containing K. If we interpret the types N and F by the objects Ne and F, respectively, in
the topos Sh(RA oKp; J) then we have, by the results proved earlier, a model of T in Sh(RA oKp; J). Let
AlgCl 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 oKp; J) with the same interpretation. Let f be a sentence in the language
such that T + AlgCl ` f in IHOL deduction system. By soundness [

This model can be implemented, e.g. in Haskell. In the paper [

? Let K be a field of characteristic 0 and G(X ;Y ) = Y n + ċin=1 bi(X )Y n i 2 K[[X ]][Y ] a monic, nonconstant polynomial separable over K((X )). Let F be the algebraic closure of K, we have a positive integer m and a factorization G(T m;Y ) = Ġin=1(Y ai) with ai 2 F[[T ]] ?

We can then extract the following computational content ? Let K be a field of characteristic 0 and G(X ;Y ) = Y n + ċin=1 bi(X )Y n i 2 K[[X ]][Y ] a monic, nonconstant polynomial separable over K((X )). Then there exist a (von Neumann) regular algebra R over K and a positive integer m such that G(T m;Y ) = Ġin=1(Y ai) with ai 2 R[[T ]] ?

For example applying the algorithm to G(X ;Y ) = Y 4 3Y 2 + XY + X 2 2 Q[X ;Y ] we get a regular algebra Q[b; c] with b2 13=36 = 0 and c2

3 = 0 and a factorization
G(X ;Y ) =
(Y + ( b
(Y + (b
Another example of a possible application of this model is as follows: suppose one want to show that
?For discrete field K, if f 2 K[X ;Y ] is smooth, i.e. 1 2 h f ; fx; fY i, then K[X ;Y ]=h f i is a Pru¨fer ring.?
To prove that a ring is Pru¨fer one needs to prove that it is arithmetical, that is 8x; y9u; v; w[yu =
vx ^ yw = (1 u)x]. Proving that K[X ;Y ]=h f i is arithmetical is easier in the case where K is algebraically
closed [

In this section we will demonstrate that in a classical metatheory one can show that the topos Sh(RA oKp; J) is boolean. In fact we will show that, in a classical metatheory, the boolean algebra structure of the subobject 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 commutative ring form a boolean algebra with the meaning of the logical operators given by : > = 1, ? = 0, e1 ^ e2 = e1e2, e1 _ e2 = e1 + e2 e1e2 and :e = 1 e. We write e1 e2 iff e1 ^ e2 = e1 and e1 _ e2 = e2

A sieve S on an object C is a set of morphisms with codomain C such that if g 2 S and cod(h) = dom(g) then gh 2 S. A cosieve is defined dually to a sieve. A sieve S is said to cover a morphism f : D ! C if f (S) = fg j cod(g) = D; f g 2 Sg contains a cover of D. Dually, a cosieve M on C is said to cover a morphism 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 2 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) is the presheaf W where for an object C of C the set W(C) is the set of closed sieves on C and for each f : D ! C we have a restriction map M 7! fh j cod(h) = D; f h 2 Mg.

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 f j dom( f ) = Rg and the minimal cosieve fR ! 0g.

Proof. Let S be a closed cosieve on R and let j : R ! A 2 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[a1; :::; an] . But then the morphism J : R ! R[a1; :::; an 1] is covered by S. Thus J 2 S since S is closed. By induction on n we get that a field automorphism h : R ! R is in S but then by composition of h with its inverse we get that 1R 2 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 W(R) is a 2-valued boolean algebra. Proof. This is a direct Corollary of Lemma 8.3. The maximal cosieve (1R) correspond to the idempotent 1 of R, that is the idempotent e such that, ker 1R = h1 ei. Similarly the cosieve fR ! 1g correspond to the idempotent 0.

Corollary 8.5. For an object A of RA K, W(A) is isomorphic to the set of idempotents of A and the Heyting algebra structure of W(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 extensions of K. Let A be an object of RA K, then A = F1 ::: Fn where Fi is a finite field extension of K. The set of idempotents of A is f(d1; :::; dn) j 1 j n; d j 2 Fj; d j = 0 or d j = 1g. But this is exactly the set W(F1) ::: W(Fn) = W(A). It is obvious that since W(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 oKp; J) is boolean.

Proof. The subobject classifier of Sh(RA oKp; J) is 1 tru!e W where for an object A of RA K one has trueA( ) = 1 2 A.

It is not possible to show that the topos Sh(RA oKp; J) is boolean in an intuitionistic metatheory as we shall demonstrate. First we recall the definition of the Limited principle of omniscience (LPO for short). Definition 8.7 (LPO). For any binary sequence a the statement 8n[a(n) = 0] _ 9n[a(n) = 1] holds.

LPO cannot be shown to hold intuitionistically. One can, nevertheless, show that it is weaker than
the law of excluded middle [

Theorem 8.8. Intuitionistically, if Sh(RA oKp; J) is boolean then LPO holds.

Proof. Let a 2 K[[X ]] be a binary sequence. By Lemma 5.7 one has an isomorphism L : F[[X ]] ! FNe . Let LK(a) = b 2 FNe(K). Assume the topos Sh(RA oKp; J) is boolean. Then one has K 8n[b (n) = 0] _ 9n[b (n) = 1]. By _ this holds only if there exist a cocover of K

fJi : K ! Ai j i 2 Ig [ fx j : K ! B j j j 2 Jg such that B j 8n[(b x j)(n) = 0] for all j 2 J and Ai 9n[(b Ji)(n) = 1] for all i 2 I. Note that at least one of I or J is nonempty since K is not covered by the empty cover.

For each i 2 I there exist a cocover fh` : Ai ! D` j ` 2 Lg of Ai such that for all ` 2 L, we have D` (b Jih`)(m) = 1 for some m 2 Ne(D`). Let m = ċt2T et nt then we have a cocover fxt : D` ! Ct = D`=h1 et i j t 2 T g such that Ct (b Jih`xt )(nt ) = 1 which implies xt h`Ji(a(nt )) = 1. For each t we can check whether a(nt ) = 1. If a(nt ) = 1 then we have witness for 9n[a(n) = 1]. Otherwise, we have a(nt ) = 0 and xt h`Ji(0) = 1. Thus the map xt h`Ji : K ! Ct from the field K cannot be injective, which leaves us with the conclusion that Ct is trivial. If for all t 2 T , Ct is trivial then D` is trivial as well. Similarly, if for every ` 2 L, D` is trivial then Ai is trivial as well. At this point one either have either (i) a natural number m such that a(m) = 1 in which case we have a witness for 9n[a(n) = 0]. Or (ii) we have shown that for all i 2 I, Ai is trivial in which case we have a cocover fx j : K ! B j j j 2 Jg such that B j 8n[(b x j)(n) = 0] for all j 2 J. Which by LC means K 8n[b (n) = 0] which by 8 means that for all arrows K ! R and elements d 2 Ne(R), R b (d) = 0. In particular for the arrow K 1!K K and every natural number m one has K b (m) = 0 which implies K a(m) = 0. By = we get that 8m 2 N[a(m) = 0]. Thus we have shown that LPO holds.

Corollary 8.9. It cannot be shown in an intuitionistic metatheory that the topos Sh(RA oKp; J) is boolean.