Prime Model Extensions for Differential Fields of Characteristic p ≠ 0

Prime Model Extensions for Differential Fields of Characteristic p ≠0Author(s): Carol WoodSource: The Journal of Symbolic Logic, Vol. 39, No. 3 (Sep., 1974), pp. 469-477Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272889 .

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THE JOURNAL OF SYbwouc LoGic Volume 39, Number 3, Sept. 1974

PRIME MODEL EXTENSIONS FOR DIFFERENTIAL FELDS OF CHARACTERISTIC p # 0

CAROL WOOD

?0. Introduction The main purpose of this paper is to show that there exists a prime differentially closed extension over each differentially perfect field. We do this in a roundabout manner by first giving new and simple axioms for the theory of differentially closed fields (in the manner of Blum [1] for characteristic 0) and by proving that this theory is the model completion of the theory of differentially perfect fields. This paper can be read independently from [10], where we gave more complicated axioms for the same theory (in the manner of Robinson [6] for characteristic 0).

I am indebted to E. R. Kolchin for answering many questions and for making the manuscript of his forthcoming book [2] available to me.

?1. Algebraic preliminaries. The basic references for the definitions and theorems in this section are [2], [4], [8], [9]. We give here only the degree of gener- ality necessary for ??2-3, and we state results from these references without proof.

DEFINITION. A differentialfield F is an algebraic field with an additional unary operation D such that, for all a and b in .F

D(a + b) = D(a) + D(b) and D(ab) = aD(b) + bD(a).

We assume that all the differential fields we consider are of the same characteristic p, where p is a nonzero prime.

The set of constants of a differential field .F forms a differential subfield %, called the constant field of -.: In general, FP c X, since D(aP) = paP~ D(a) = 0 for any a e -.

DEFINITION. A differentially perfect field -F is a differential field such that yp = W.

Two facts about differentially perfect fields which are of importance to us are the following:

LEMMA 1. For any differential field .F there is an extension F' of JF such that .F' is differentially perfect.

PROOF. See either [2] or Theorem 2 of [10]. LEMMA 2. Differentially perfect fields have the amalgamation property; i.e., if

JJ nJ 2 are differentially perfect fields such that .F c -F, and -F a F2 then there exists a differentially perfect field F3 such that JF a .SF3 and .2 c I3.

PROOF. See Seidenberg [8]. DEFINITION. Let F and *% be differential fields, with -F c *. Then *' is

separable over JF provided Y and ,%Z' are linearly disjoint over FP.

Received June 24, 1973.

469 C 1974, Association for Symbolic Logic



470 CAROL WOOD

Proposition 1 of Chapter II, ?2 of [2] says it is actually sufficient, for V' to be separable over Y that W and *P be linearly disjoint over FP, where W is the constant field of .: It follows that every extension of - is separable if and only if - is differentially perfect.

For an arbitrary index set I, let F{Yi}ie1 denote the differential polynomial ring over Y in the differential indeterminates {Yi}ici; this is the set of all field theoretic polynomials in indeterminates {yi, Dy1, D2y., ... }ieC,, where D'y1 is thejth derivative of the indeterminate yi, for any positive integer j. Let SF c Y and {7)i}ie? c X.

Then SK7qiei denotes the differential subfield of M% generated by {Qq}1Jj; this is the same as the field extension L(hqj, D,, D2, *, with the definition of D given as the restriction of D on *'.

We now restrict our attention to finite I. Given an n-tuple (-ql, - - *, r7n) of elements in some extension of a differential field - there exists an ideal 91 in fl{y1, - - -, yJ.

called the defining differential ideal of (7ql, - - *, r7n), which consists of those differential polynomials of flyl, - - , yn} which vanish at (iq, - - , r7n). If 9I # 0, then

(771*, . . n) is called differentially algebraic over - When F<771, * * , 77n> is separable over - the finite basis theorem (see [2, Chapter III, Theorem 1]) says that 9 is determined by a finite subset, - *-* ,ft of 91, in the sense that any zero offI,, - - X.f in any extension of J must also be a zero of every differential polynomial in 91. If the n-tuple (n,* , J) has a defining differential ideal .9 in F{y,, - - *, yJ and 9D c .2 then (n,..J, n) is a (differential) specialization of (7,1, 7. D) over A The specialization is called generic if 9 = .9, in which case !<711, 7*A> and F<;>1 * * s Cn> are isomorphic over , under the map sending 7i to h, i = 1,* , n.

DEFINITION. Let 7.1-, 7i,, be elements of some extension of A. The n-tuple (71 ---, 7 .) is constrained over J provided

(M) < * * ,> is separable over r and (2) there exists B(y1, . * , Yn) e y * yXJ} such that B(,1q, - - 7,n) # 0 and

such that if (n1, * *, ,n) is a nongeneric specialization of (7 1, 7 *,) with

SF\ 1 . * ) tn> separable over -, then B(41, - - *, An) = 0. Any B satisfying (2) is called a constraint of (711,- - **, 7un) over F. Condition (2)

says that the defining differential ideal of (711, - *, 77n) is maximal with respect to exclusion of B. By the finite basis theorem there exist f, , f- e{yl- --yn} such that fl(,ql, - - *, en) = f2( 1,

- - ', 7n) =. = fs(i1 - -, X n) = 0 and such that

for any n-tuple (n,.*1 , - ) with fl(4,, - - ) = * = fs(;,s - - n) = 0 and B(41, - -, i) # 0 it must be that -<771, *, On> and -F<1, * * n> are isomorphic over F. The fact that the isomorphism type of a constrained n-tuple over -F is completely determined by a finite number of differential polynomial equations and one inequation is exactly what is needed from a model-theoretic point of view, as we see later. The rest of this section contains information about extensions of differential fields which we shall apply to the constrained case. As before, we consider only finitely generated extensions.

THEOREM 3 (KOLCHIN [2, PROPOSITION 6, CHAPTER III, ?10]). Let JF<71, *,n> be separable over IF, and let B(yl, * * *, YJ) e -F{Yl, * * *, yn} be such that B(771,, *, * *n)

# 0. Then there exists a specialization (41, ..., in) of (771 ...* * Xn) such that

(41n . * CO) is constrained over - with constraint B. THEOREM 4 (KOLCHIN [2, PROPOSITION 7, CHAPTER III, ?10]). Let 771, * 7 - n,



PRIME MODEL EXTENSIONS FOR DIFERENTIAL FIELDS 471

419 * * Ck be elements of some extension of the differential field F and let W be the constant field of .

(a) If .F<71,*, m7> = JK<il * k> and if the n-tuple (rj, *,n) is constrained over Y, then the k-tuple (ul,* , ok) is also constrained over -

(b) If F<71, - - *, cns , 41 - -

* 4k> is separable over F<77, * - , 7n> and the (n + k)- tuple (77, r * *, E 41 * * * Ck) is constrained over i; then (n, .., k) is constrained over F<771, - * *, 70i> and (7713, * * *, an) is constrained over -F.

(c) Suppose the constantfield of ;<771,.- -, 3n> is separable over (JF<K71, *-* * 7>)Pce.

If (41, . . . k) is constrained over F<711, - - *> and (1, ***, 77) is constrained over

-F. then (77 *1 *, ns . . . * Ck) is constrained over -

(d) If (ri, * *, C7n) is constrained over -F then the field of constants of Y<7711, *, *7n> is separably algebraic over ((F<7, * * ., X/7>)Pe.

COROLLARY 4.1. Let rj1, - - ., 77n3 C13 * * k be elements of some extension of - where sF is differentially perfect.

(c') If (E13, - - , W) is constrained over -F<771,, *, rn> and (771X,..., **n) is constrained over , then (h,.*1 , -, Ck 771 , On) is constrained over E

(d') If (771, .. *,7an) is constrained over i, then JF<771, - *, 7in> is differentially perfect.

PROOF. By (d) of Theorem 4, the field of constants of F<71,*, 77n> is separably algebraic over r<71, *, '7n>)PW But

05F<771, * * * qn>)Pef = P'7<7713, ', *77 n>) s

since W = 5P; an element of *, 7wn> which is separably algebraic over

F<7712 . . .

9 7>)p must lie in (-F<nl, * - *, 7sn>)P- Thus the constant field of

Y<13, * *, *7n> is ( *1,

, * *

i X>)', and (d') is proved.

Also, since the constant field of <71, - - *, 7n> is (.F<7*1,---, * n>)P whenever

(771 '

* * n) is constrained over J we can drop the additional hypothesis of (c) to obtain (c').

It is often desirable to consider only simple extensions; we describe next the conditions under which we may assume any differentiably algebraic extension is simple.

THEOREM 5 (SEIDENBERG [9]). Let -F be a differentialfield with constant field W such that -F has infinite linear dimension over W. If 7h13, - *, 7, are differentially algebraic over , then there exists 4 such that YF<771, - ,- rn> = at<;>.

We remark that the existence of infinitely many linearly independent elements in . over W guarantees that any nontrivial differential polynomial inequalities over JF are satisfiable in - In characteristic 0 (but not in general) a single nonconstant element suffices to give F infinite dimension over W.

DEFINITION. Let n be a positive integer. The nth order Wronskian w7, is the differential polynomial given by the determinant

Y1 *-- Yn

Dyl ... DYn

Wn(Y13 *, * ) A D) y ... Dyn.

Dn - 'y ... Dn-lyn



472 CAROL WOOD

THEOREM 6 (Rirr [4, P. 34]). The elements 1 n E, e are linearly independent over the constant field le if and only if w^(77, * n) : 0.

Ritt's proof for characteristic 0 is valid for arbitrary characteristic. In particular, Theorem 6 shows that linear independence over constants is invariant under extension of the field S

COROLLARY 6.1. If'i ,, ** *,, e- -S are such that

D7in = 03, D"-71i 0 0 for i =1***,n,

then 7ll, .., r7n are linearly independent over W.

PROOF. By looking at the Wronskian, we see that

771 712 ... S9n

o D7)2 D77n Wn()13 *, -9n)

= 0 0 D293 .... D277n = 7(D72)(D2713) ... (Dn- 1n) 96 0.

0 0 ... 0 Dn-1. 7n

Thus 71, - - , 77n are linearly independent over W by Theorem 6. COROLLARY 6.2. Let S be differentially perfect. For each positive integer n

there exists 7wn constrained over S such that SF<7)n>n= 1,2 has an infinite linear basis over its constant field.

PROOF. A solution for Dny = 0, Dn- ly # 0 must exist over any differential field and is obtainable by adjoining n algebraically independent elements to,, t1, to the field and assigning

Dtj-l = t,, j = 1, ,n-1, and Dtn1 = O?

Thus by Theorem 3 there exists a constrained specialization in, of such an extension of S with constraint Dn -'y. By Corollary 6.1 the elements 771, 12. are linearly independent over W, hence over the constant field of -F<'In>n= 12, -

By combining Theorem 5 and Corollary 6.2 we can extend any differentially perfect field, via the adjunction of certain constrained elements, to one over which any finitely generated differentially algebraic extension is primitive. For any such primitive extension K<C> of a differentially perfect field f C is a generic zero of some irreducible differential polynomial f(y) e S{y}. The isomorphism class of SF<> over S is completely determined by the equation f(y) = 0 together with all inequations of the form g(y) # 0 where g(y) e f{y} and g(y) has order less than the order of f(y). (The order of an element of F{y} - S is the largest n such that Dny occurs in it nontrivially; if no Dny occurs, the order is 0, and we assign order -1 to nonzero elements of -S.) If C is constrained over S with constraint B(y), then the isomorphism class of ; over S is determined by f(y) = 0 and B(y) # 0. The constraint B may also be chosen to have order less than that off(y). (In casef(y) is of order 0, then f(y) is an irreducible algebraic polynomial, and B(y) = I will do.) To replace a given B(y) by a constraint of order less than f(y), we can either argue model-theoretically via compactness, or can reduce B(y) modulo the ideal generated byf(y) to a polynomial of order at most the order of f(y), as in Ritt [4, p. 6]. Since f(y) is irreducible, we can look at the reduction process and pick out a polynomial of order strictly less than that off(y) which is



PRIME MODEL EXIENSIONS FOR DIFFRENTIAL FIEDS 473

nonzero on exactly those zeros of f(y) for which B(y) is nonzero. Therefore a simple constrained extension of a differentially perfect field is determined by an equation f(y) = 0 and an inequation g(y) : 0, where the order of g(y) is less than that off(y).

Conversely, given such a pair of polynomialsf(y) and g(y) in F{y}, where .F is differentially perfect and the order off(y) is greater than that of g(y), there must exist an extension of -F in which f(y) = 0, g(y) # 0 has a solution. This is true since f(y) must have an irreducible factor fo(y) of the same order as f(y), and a generic zero of fo(y) will satisfy f(y) = 0, g(y) # 0; a generic zero of fo(y) is found by taking a field extension F(to0,* *, tj) of .F where to0,**, t,- 1 are algebraically independent over Y and where to is a zero of the polynomial

obtained by replacing Dly by to, i = 0O... , n - 1, and Dhy by x. The assignment Dti = tj + , i = 0, . - *, n, uniquely determines the differential field extension Y<t0> of ., and to is a generic zero off0; the fact that .F is differentially perfect guarantees that fo(y) is separable.

?2. Differentially closed fields. Now we describe the model-theoretic setting for differential fields of characteristic p. First, let L be the language of fields (two constants, 0 and 1; two unary functions - and -1; two binary functions + and *) together with one additional unary function D. The language L is the extension of L obtained by adding another unary function r. The axioms for differential fields of characteristic p are the axioms for fields of characteristic p plus two axioms:

(*i) VxVy(D(x + y) = D(x) + D(y)) and

(**) VxVy(D(x-y) = x D(y) + y- D(x)).

The theory TY of differentially perfect fields is the union of T and the axiom

Vx((D(x) = 0 A (r(x))P = x) v (D(x) # 0 A r(x) = 0)).

The theory T is actually a theory in L, but T is not; we could replace (***) by

Vx~y(D(x) = 0 = yP = x)

and obtain T' as the theory of differentially perfect fields in L, but the fact that T' is not universal in L (and Ti is universal in Z) would cause us difficulties later. As usual, we denote by L(Y) the language obtained from a structure JF for a language L by adjoining a new constant for each element of -. The diagram 2(Y) is the set of all atomic and negated atomic formulas of L(.F) satisfied by F.i

One awkwardness resulting from the use of L is that a model-theoretical simple extension of a model of Ti differs in general from a differentially algebraic simple extension of a differentially perfect field, since the latter may introduce constants without pth roots. By Corollary 4. 1(d), these two kinds of extensions do correspond if the element adjoined is constrained. Another awkwardness is that an existential sentence q' in L is equivalent to the disjunction of a number of conjunctions of differential polynomial equations and inequations, while an existential sentence



474 CAROL WOOD

w in L may contain occurrences of r. However, by writing p as a disjunct of con- juncts and replacing each term of the form r(t) by a new variable x, adding the sentence xP = t to the conjunct in which t occurs, and putting 2x in front of the original quantifiers we can find an existential sentence p of L such that

T F wa

We shall move freely back and forth from p to ap, taking into account the additional quantifiers in p.

Now we define a third theory T*, the theory of differentially closed fields. For each positive integer n, let n,, be the sentence in L (in fact, p,, is in L) which states that there exists a solution to f(x) = 0, g(x) # 0 for every pair of differential polynomials in one differential indeterminate such that f and g have order and total degree at most n, and such that f has order greater than g. Let T* = TU {on I n is a positive integer}.

LEMMA 7. The theory T* is model consistent relative to T; i.e., any differentially perfect field can be extended to a differentially closedfield.

PROOF. The proof is the usual chaining procedure, using Lemmas 1 and 2 to guarantee that we can continue to adjoin solutions to equations and inequations. Given F T, there exist J k T such that JF = So, J c A + 1, i = O. 1, * , and such that any pair, g as in ,, with coefficients in Aj has a solution in Ai+1. Then 9 = U{9} = l,... is a model of T* extending s

LEMMA 8. Let JF T-* and let the n-tuple (r1, -, qn) be constrained over Then 771, -, 71 are elements of -

PROOF. First let n = 1; 91. = 9. By the closing remarks of ?1 there is a pair f(y), g(y) E J{y} with order off greater than that of g such thatf(l7) = 0, g(,q) # 0 and such that if f(g) = 0 and g(4) + 0 for any g in any extension of - then 9K7> and J<K> are isomorphic over -i Since F k t*, there must exist g E -F such

that f(4) = 0, g(4) # 0. Thus F<71> and F<4> = F are isomorphic over i; i.e.,

Now by Corollary 6.2 there exist {}i = 1,2... with each Of constrained over - such that <g, =1,2,... has an infinite linear basis over its constant field. By the above proof for n = I and = hi, F<4i>i= 1,2 .. = .Fi If (71D, , TO) is constrained over -; then F<'91, , On> is a differentially algebraic extension of - and therefore there exists q E F such that -r<71, , TO>= JF<77>, by Theorem 5. By Theorem 4(a), 7 is constrained over .; therefore n E J and <711, ,q> =

as desired. We have chosen to call T* the theory of differentially closed fields, in analogy

with the characteristic 0 theory of Robinson [6]. It is however true that proper differentially algebraic extensions of models of T* exist. The above lemma shows that no proper extension via a constrained element exists, and an alternative name for T* is the theory of constrainedly closed differential fields, as suggested by Kolchin. In our opinion, the name "constrainedly closed" has the drawback of saying too little, since a constrainedly closed differential field does have solutions to all consistent finite systems of differential equations and inequations (and not just constrained ones, as a superficial reading of the name suggests), while " differentially closed " claims too much. A third option is " differentially complete,"



PRIME MODEL EXTENSIONS FOR DIFFERENTIAL FIELDS 475

in keeping with the notion of an existentially complete structure, which is currently in use in model theory. We offer no excuse for ignoring this option other than our own bad habits.

TBEoREm'9. The theory T* is the model completion of T. PROOF. In Lemma 7 we established the model consistency of T* relative to T.

Since P has the amalgamation property (Lemma 2) it is sufficient to show that T* is model complete. By Robinson's test [6] we need only show that any model F of T* is existentially closed in any extension A' 2 F such that JF' F T. By the argument given earlier, an existential sentence in L(.F) is equivalent to one in L(s) (here we are regarding JV as a structure for both), and so it is enough to show that givenf1(yl, ** , yn),. Y-).o(. * *' , Y), g(Y. , Y y.n) e .F{yl * yj} and elements 11,--, * , e Y' such that f'(X71, -.., 77) -- = f*(* , .*., *n) = 0,

g(71,, * * 0 that there already exists a solution to fi =. f, = 0, g # 0 in -Fi By Theorem 3 there exists a constrained specialization (,. * , On) of (7/113,

. . / 77) with constraint g in some extension of J By Lemma 8, n,* *, n e t

and the proof is finished. COROLLARY 9.1. The theory T* is complete and substructure complete. PROOF. P has a prime model, the constant field .9; with p elements. Therefore

7* is equivalent to T~* u , which is complete by Theorem 9. Since T is universal, for any substructure F of a model of T* we have - F T; hence T* u 2(.) is complete, again by Theorem 9. Thus T~* is substructure complete.

We observe that the model completion T* of T' in L is thus obtainable by replacing T by T' in the definition of T*, since the only occurrence of the function r in T* is in axiom (***) of T. This theory T* is the model companion of T; T* is also complete by the same argument as for T*, but T* is not substructure complete.

?3. Morley analysis of Tl*. DEFINITION. Let K be a complete theory. An n-type of K is a set q of formulas

in the language of K with n free variables x1, x,, such that the following hold: (1) q is consistent with K; i.e., if p1,--., p*, e q, then K F 3x ... 3X,,(4p A ... A 9's). (2) q is maximal; i.e., if 'p is a formula of the language of K, with free variables

x1,-- , x,, then either 'p eq or (-np) eq. In particular, if-s' F Kand a,, *a,, e- Es, then {g(xl, *,) | F p(al,*, a,,)}

is an n-type, called the n-type realized by (a,, - - *, a,,) in d. An n-type q is principal if there is an element p e q (called a generator of q) such

that, for all b e q,

K 1 Vxli**. Vx,,('p '

We shall be interested in complete theories of the form t" U 9(f) in the language Z(s), where J> F T; an n-type of T~* u 9() will be called an n-type over -. For a fixed n > 0, the set of all n-types over F forms a topological space S,,(S.), under the topology with basic open sets of the form U0 = {q I q is an n-type and 'p e q}, where p ranges over all formulas in L(.F) with free variables xi, , x,. The space S,,(.F) is compact Hausdorff with clopen subsets as a base, i.e., Sj(Y) is a Stone space. Since T* is substructure complete, every formula is equivalent to a quantifier-free formula with respect to T*, and so every q e S4(F) is determined



476 CAROL WOOD

by the quantifier-free formulas in q. The principal n-types are exactly the isolated points of Sn(J); if 'p generates q, then

U. = {q}.

LEMMA 1 0. Let F k T and let (,11, * * ,rn) be constrained over -i Then the n-type realized by (r, - -, 71n) over J is principal.

PROOF. The n-type realized by (71, - -, 7in) over F is the n-type of (X1,* rqn) in T* u 9(.); this is unambiguous since T~* is substructure complete. By Corollary 4.1 (d') the extension <7K1, - - n> is differentially perfect, so the model-theoretic extension of F by 71,.. , 7n is just F<71, ,7n>. The n-type of (%), --, 7) over - therefore corresponds to the differential-algebraic isomorphism class of

(i*1,Bt * n) over -i Since (771, , 7n) is constrained over -F there exist fl, -.- and BEY{y1, .,Y} such that the conditions f1 = 0, B # 0 are satisfied by (X1, *.9n) and completely determine the isomorphism class of

(77 77 n) over 3 Thus the n-type of (791, n, n) over Y is principal and is generated by the logical formula (in L actually) in the variables x1, **, xn which says that fl(xl, -, xn) = = f(xl, -, xn) = 0, B(x1, -, Xn) # 0.

THEOREM 11. Let F k T. Then the isolated points in S1(.f) are dense in S1().%f PROOF. Let U. be a nonempty basic open set in S1(.), where 9 has one free

variable x1. Since U0, + z, T* u .(F) F 3x1p. We may assume 9 is quantifier- free, since T* is substructure complete. By our procedure of translating into L, we know there exists a formula '- = p'(x1, x2, **,Xk) in L such that Tu 9(F) F 9 - (3x2.. 3Xk)JP'. By the usual argument, then, we can reduce the problem to where ap' is a system of differential polynomial equations and inequations in variables x1, -*, Xk of the form f1(xl, -., xk) = = fs(x1,. * ,Xk) = O, g(x1,". *, Xk)

9 0. Since 'p is consistent, there must exist a constrained k-tuple (771, , Ok) in some extension of Y with constraint g(Y1, - * , Yk) such that f1(X1, *, 'qk) = *

= fSs(77 , 7k) = 0. By Lemma 10, the k-type of (71, -, 9k) is principal, generated, say, by b'(x1, ..., Xk). Thus U*, c U., as open sets in Sk(sF). Now translate /' back into L by introducing r whenever necessary, so that the resulting b has only one free variable x1. Then b generates a principal 1-type over F and

U,, c U,, as desired. Thus the isolated points of S1(Y) are dense. DEFINITION. Let K be a complete and substructure complete theory, and let

K. be the universal part of K. If -sa' c a, V Kv,, then .v is a prime model extension of -V provided

(1) WOK, (2) if c a W and W h K, then there exists an isomorphism of Y into W over . Morley (Theorem 4.3 of [3]) is proved, for K as in the above definition, that if

the isolated points of S1(sl) are dense in S1(01) for every dV Kv, then every V K, has a prime model extension.

COROLLARY I 1. 1. Over every J F T there exists a prime model extension. PROOF. Immediate from Theorem 11 and the above result of Morley. Corollary

1 1.1 has been obtained independently by Shelah [11]. This corollary gives some justification for talking about a differential closure

(or a constrained closure) of a differentially perfect field. It would be of interest, therefore, to know whether or not a prime model is unique.



PRIME MODEL EXTENSIONS FOR DIFFERENTIAL FIELDS 477

We observe that a sort of converse of Lemma 10 holds: Given a principal n- type q over F, F T. the corresponding differential field extension by a realization of that type will be of the form - , 7s> where (qj, -, - ) is constrained over F but where s may be larger than, n; this can happen if a generator of q contains occurrences of the function r.

REFERENCES

(1] L. BLUM, Thesis, M.I.T., Cambridge, Mass., 1968. [2] E. R. KOLCHIN, Differential algebra and algebraic groups, Academic Press, New York,

1973. [3] M. MORLEY, Categoricity in power, Transacions of the American Mathematical Society,

vol. 114 (1965), pp. 514-538. [4] J. Rrrr, Differential algebra, Dover, New York-, 1966. [5] A. ROBINSON, An introduction to model theory, North-Holland, Amsterdam, 1965. [6] , On the concept of a differentially closed field, Bulletin of the Research Council of

Israel, Section F, vol. 8F (1959), pp. 113-128. [7] G. SACKS, Saturated model theory, Benjamin, Reading, Mass., 1972. [8] A. SEMDENBERO, An elimination theory for differential algebra, University of California

Mathematics Publications, vol. 3 (1956), pp. 31-65. [91 , Some basic theorems in differential algebra, Transactions of the American Mathe-

matical Society, vol. 73 (1952), pp. 174-190. [10] C. WOOD, The model theory of differential fields of characteristic p # 0, Proceedings of

the American Mathematical Society, vol. 40 (1973), pP. 577-584. [11] S. SHELAH, Differentially closed fields (preprint).

YALE UNIVERSIY

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Prime Model Extensions for Differential Fields of Characteristic p ≠ 0

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