Liouville's Theorem on Integration in Terms of Elementary Functions

Liouville's Theorem on Integration in Terms of Elementary Functions R.C. Churchill Prepared for the Kolchin Seminar on Di®erential Algebra (KSDA) Department of Mathematics, Hunter College and the Graduate Center, CUNY September, 2006 (This is a corrected and extended version of a lecture originally given in September 2002.) This talk should be regarded as an elementary introduction to di®erential algebra. It culminates in a purely algebraic proof, due to M: Rosenlicht [Ros2], of an 1835 theorem of Liouville on the existence of \elementary" integrals of \elementary" functions. The precise meaning of elementary will be speci¯ed. AsR an application of that theorem we prove that the inde¯nite integral ex2 dx cannot be expressed in terms of elementary functions. Contents x1. Basic (Ordinary) Di®erential Algebra x2. Di®erential Ring Extensions with No New Constants x3. Extending Derivations x4. Logarithmic Di®erentiation x5. Integration in Finite Terms References 1 Throughout the notes \ring" means \commutative ring with (multiplicative) identity". That identity is denoted by 1 when the ring should be clear from context; by 1R when this may not be the case and the ring is denoted R. The product rs of ring elements r and s is occasionally denoted r ¢ s. 1. Basic (Ordinary) Di®erential Algebra Throughout the section R is a ring. An additive group homomorphism ± : r 2 R 7! r 0 2 R is a derivation if the Leibniz rule (1.1) (rs) 0 = r ¢ s 0 + r 0 ¢ s holds for all r; s 2 R ; when ± is understood one simply refers to \the derivation r 7! r 0 (on R)". The kernel of a derivation refers to the kernel of the underlying group homomorphism, i.e., f r 2 R : r 0 = 0 g. Note from (1.1) and induction that (1.2) (rn) 0 = nrn¡1 ¢ r 0; 1 · n 2 Z: 1 A di®erential ring consists of a ring R and a derivation ±R : R ! R. When (R; ±R) and (S; ±S) are such a ring homomorphism ' : R ! S is a morphism of di®erential rings when ' commutes with the derivations, i.e., when (1.3) ±S ± ' = ' ± ±R : Di®erential rings constitute the objects of a category; morphisms of di®erential rings form the morphisms thereof. An ideal i of a di®erential ring R is a di®erential ideal if i is \closed under the derivation", i.e., if r 2 i implies r 0 2 i. The kernel of any morphism in the category of di®erential rings is such an ideal. The quotient of a di®erential ring by a di®erential ideal inherits a derivation in the expected way, and the anticipated factorizations and isomorphisms are valid. In particular, when f : R ! S is a morphism with kernel i and ' : R ! R=i is the canonical homomorphism there is a factorization f R ¡! S ¼ (1.4) ' # % R=i 1More generally, one can consider rings with many derivations. In that context what we have called a di®erential ring would be called an \ordinary di®erential ring". We have no need for the added generality. 2 into morphisms of the category. When (R; ±R) is a di®erential ring and r 2 R one often refers to the evaluation ±Rr as \di®erentiating r". When r; s 2 R satisfy ±Rr = s one refers to s as the derivative of r and to r as a primitive2 of s. Examples 1.5 : (a) When x denotes a single indeterminate3 over R the usual derivative d=dx gives the polynomial algebra R[x] the structure of a di®erential ring. More generally, when R[x] is the polynomial algebra over R in a single indeterminate x the P j P j¡1 mapping d=dx : j rjx 7! j jrjx is a derivation on R[x], and thereby endows R[x] with the structure of a di®erential ring. This is the usual derivation on R[x]. (b) The mapping r 2 R 7! 0 2 R is a derivation on R, called the trivial derivation . (c) Let F be a ¯eld, let x be a single indeterminate over F , and give R := F [x] the usual derivation. Then the only proper di®erential ideal of R is the zero ideal. This follows immediately from the PID property of R and the fact that the derivative of any generator of a non-zero ideal, being of lower degree than that of the generator, cannot be in the ideal. (d) The only derivation on a ¯nite ¯eld is the trivial derivation. Indeed, when K is a ¯nite ¯eld of characteristic p > 0 the Frobenius mapping k 2 K 7! kp 2 K is an isomorphism, and as a result any k 2 K has the form k = `p for some ` 2 K. By (1.2) we then have k 0 = (`p) 0 = p`p¡1` 0 = 0, and the assertion follows. The usual derivation on the polynomial ring R[x] is certainly familiar from elementary calculus, where it is primarily used for the analysis of functions. However, readers should also be aware of the fact that this derivative can be useful for purely algebraic reasons, e.g., for investigating the multiplicity of roots. To recall this application let K be a ¯eld, let p 2 K[x] be a polynomial, and let ® 2 K be a root of p. Then we can write p(x) = (x ¡ ®)mq(x) ; 2One is tempted to refer to primitives as \anti-derivatives" or \inde¯nite integrals," but that terminology is seldom encountered. 3 As opposed to an n-tuple x = (x1; : : : ; xn) of algebraically independent elements over R. 3 where q 2 K[x] is relatively prime to x ¡ ® and m ¸ 1 is an integer. When m = 1 the root is simple; otherwise it is multiple. In either case m is the multiplicity of the root. Applying the usual derivation to the displayed formula we see that p 0(x) = (x ¡ ®)mq 0(x) + m(x ¡ ®)m¡1q(x) ; and therefore p 0(x) = (x ¡ ®)q 0(x) + q(x) when m = 1 ; ¡ ¢ p 0(x) = (x ¡ ®)m¡1 (x ¡ ®)q 0(x) + mq(x) when m > 1: Since q(®) =6 0 we have the following immediate consequence4. Proposition 1.6 : Let K be a ¯eld, let x be a single indeterminate over K, assume the usual derivation on K[x], and suppose p 2 K[x]. Then an element ® 2 K is a multiple root of p if and only if p(®) = p 0(®) = 0. Corollary 1.7 : Suppose L ⊃ K is an extension of ¯elds and p 2 K[x] is such that p and p 0 are relatively prime. Then p has no multiple root in L. Proof : From the relatively prime assumption we can ¯nd elements g; h 2 K[x] such that gp + hp 0 = 1, and if ` 2 L were a multiple root then evaluating at ` would yield the contradiction 0 = g(`) ¢ 0 + h(`) ¢ 0 = g(`)p(`) + h(`)p 0(`) = 1. q:e:d: Corollary 1.8 : Suppose L ⊃ K is an extension of ¯elds and p 2 K[x] is irreducible. Then p has no multiple root in L if and only if p 0 6= 0. Proof : ) If p 0 = 0 and ` 2 L is a root of p then ` must be a multiple root by Proposition 1.6. ( Since p is irreducible p 0 has lower degree than p the two elements p and p 0 must be relatively prime. We conclude from Corollary 1.7 that p has no multiple root in L. q:e:d: We need a characteristic zero consequence of Proposition 1.6 which requires (a) of the following preliminary result. Assertion (b) is never used in our subsequent work, but contrasting the 0 characteristic and positive characteristic cases can be instructive. 4The presentation of this result, and the two corollaries that follow, is patterned after [Hun, Chapter III, x6, Theorem 6.10, pp: 161-2]. 4 Proposition 1.9 : Suppose K is a ¯eld, x is a single indeterminate over K, and p(x) 2 K[x] has degree n ¸ 1. Assume the usual derivation on K[x]. Then the following statements hold. (a) When char(K) = 0 the usual derivative p 0(x) of p(x) has degree n ¡ 1. In particular, p 0(x) 6= 0. (b) When char(K) = p > 0 the polynomial p(x) satis¯es p 0(x) = 0 if and only if Pn j p(x) has the form p(x) = j=0 kjx , where pjj whenever 0 6= kj 2 K. Pn j 0 Pn j¡1 Proof : We have p(x) = j=0 kjx and p (x) = j=0 jkjx , where n > 0 and kn 6= 0. 0 (a) If char(K) = 0 then nkn 6= 0, hence p (x) 6= 0. 0 (b) If char(K) = p > 0 then p (x) = 0 if and only if jkj = 0 for j = 1; : : : ; n, i.e., if and only if jkj ´ 0 (mod p) for j = 1; : : : ; n, and this is obviously the case if and only if pjj whenever 0 6= kj 2 K. q:e:d: Theorem 1.10 : Any algebraic extension of a ¯eld of characteristic 0 is separable. Proof : By Proposition 1.9(a) and Corollary 1.7. q:e:d: Remark 1.11 : Theorem 1.10 is false when the characteristic 0 assumption is dropped. To see ap speci¯c example let t be an indeterminate over Z=2Z, let K = (Z=2Z)(t) and let t 2 Ka be a root of p(x) = x2 ¡t 2 K[x].

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