View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 28, 88-101 (1974) Prime and Composite Polynomials F. DOREY George hfason University, Fairfax, Virginia 22030 G. WHAPLES Umversity of Massachusetts, Amherst, Massachusetts 01002 Communicated by D. A. Buchsbawn Received April 10, 1972 1. INTRODUCTION If f = f(x) and g = g(x) are polynomials with coefficients in a field A,, then f 0 g shall denote the polynomialf(g(x)). The set k,[.~] of polynomials is an associative monoid under this operation; the linear polynomials are the units; one defines primes (= indecomposable polynomials), prime factoriza- tions (= maximal decompositions) and associated primes and equivalent decompositions just as in any noncommutative monoid. In 1922 J. F. Ritt [9] proved fundamental theorems about such polynomial decompositions, which we restate here in Section 2. He took k, to be the complex field and used the language of Riemann surfaces. In 1941 and 1942, H. T. Engstrom [4] and Howard Levi [7], by different methods, showed that these results hold over an arbitrary field of characteristic 0. We show here that, contrary to appearances, Ritt’s original proof of Theorem 3 does not make any essential use of the topological manifold structure of the Riemann surface; it consists of combinatorial arguments about extensions of primes to a composite of fields, and depends on the fact that the completion of a field k,(t), at each of its prime spots, is quasifinite when k, is algebraically closed of characteristic 0. Our Lemma 1 below contains all the basic information which Ritt gets by use of Riemann surfaces. Since his methods are extremely interesting and probably have further useful applications we give in Section 3 a simplified version of his proof in the language of valuation theory. In Section 4 we discuss polynomial decompositions over a field of charac- teristic p + 0, give counter-examples for several results which hold in characteristic 0, and prove that every decomposition of an additive polynomial is equivalent to a decomposition into additive polynomials. 88 Copyright Q 1974 by ,4cademic Press, Inc. All rights of reproduction in any form reserved. PRIME AND COMPOSITE POLYNOhlIALS 89 2. RITT'S RESULTS Let k, be any algebraically closed field of characteristic 0. In Theorem I, Ritt used the “monodromy group off-‘(x)” instead of the one we mention; it is easy to show that these are equivalent permutation groups. THEOREM 1. Let f(x) E kO[x], let u = f(x), and let G be the Gulois gro+ of the splitting field D off(X) - w oaep k,,(w). Then f(x) is indecomposable in k,[x] e the representation of G as a group qf permutations of the Toots of f(X) - w = 0 is a pGnitive permutation group. THEOREM 2. Let f(x) E k,[x] and Zet be maximal decom$ositions. Then T = s. The decompositions (1) and (2) are equivalent o degree fi(x) = degree gi(x) f OY each i. The sequence (degree gl) is a permutation of (degree fi}. And one can get from (I) to (2) by a chain of maximal decompositions saclz that at each step onl$ two adjacent conapoT:ents are ch.azged. DEFINITION 1. Call an identity fi o fz = g, o g, a bidecom9ositioz if theji I gi are indecomposable and fi is not an associate of g, . (We say j and g are associates of = Gr 0 g 0 [a where the ki are linear.) Call two bidecompositions equivalent if corresponding components in them are associates or if this is so after interchanging the two sides of one of them. Call a bidecomposition exponential if it is equivalent to one of the form and trigonometric if it is equivalent to one of form t&g 0 t,,(x) = t&z) 0 t&v), (4) where the polynomials t,,(x) are defined by the identities (2” + x-m)/2 = fm((z + .+)/2j (5) or cos 7n.x = t,(cos x). 16) THEOREM 3. Every bidecomposition is either exponential OY trigonometric. 90 DOREY AND WJJAPLES 3. PROOFS Let K, be any field and t an element which is algebraically independent of k, . If k = k,(t) let Jl(k) denote the set of all prime spots of k; i.e., the set of all equivalence classes of valuations of iz which are trivial on k, or, equiva- lently, the set of equivalence classes of places of k into k~lg”‘os u {mf (see [I]), and let p(a) denote the prime spot corresponding to the valuation with 1t 1 > 1, or the place with t -j co. Of course p(a) depends on our choice of the generator t of k. Let K = k,( x7) wTh ere K, is algebraically closed. By Luroth’s theorem (see van der Waerden’s “Modern Algebra,” Vol. I, for an elementary proof) every subfield L contained in K and different from k, is of form L = k,(w) where w E K is a rational function of X. If p E M(L) and p has extensions P, E M(K), with ramification numbers e(l),..., e(r) we write as usual ;e;'i;]) p = p;u,p;ca ... P:Ir). It is easy to check that a subfield L of K,(X) is of form k,(w) with w a polynomial in i, if and only if the prime P(W) of 12,(x) is totally ramified in K/L; i.e., there is a prime p(a) E flf(L> with p(m) = P”(a). Supposef = f(x) h as a decomposition (I). Then there is a chain of fields k,,(x) 3 Kl 3 K, --- 3 K, = k,(w), (7) Ki = ko(fi(fi-1 ‘-- U&4 ->)s where fJJ=f(x); (8) degreeKg/K,+, = degreefi and P(W) is totally ramified in K/K, . Conversely, supposew = j(x) is a polynomial and me have a chain of fields (7). It is well known that every k,-automorphism of k,(t) maps t into some element (at + @l(ct + 4, and every k,-automorphism which leaves the prime p(a) fixed maps t into an element f(t) = at + b. Using these facts it is easy to show that from any chain of fields (7) in which P(W) is totally ramified we can produce a sequenceof polynomialsfi satisfying (8) and (l), and that any two decompositionsoff(x) d erived in this way from the samechain of fields (7) are equivalent, Unfortunately it does not, in general, work in the other direction; in Section 4 we shall give an example of two equivalent decom- positions which give different chains of fields. However it follows from Theorem 2 that if k,, is algebraically closed of characteristic 0 then there is a one-one correspondencebetween equivalence classesof decompositions(1) and chains of fields (7). In the situation described in Theorem 1 let K = k,,(x) be the fixed point field of the subgroup H C G = Galois group of Q/k. From the above discussionit follows thatf is indecomposableo there is no group H’, different from G and H, with H C H’ C G. It is a well-known theorem of group theory that this is so o the representation of G as a group of permutations of the PRIME AND COR~IFOSITE POLYNOhIIALS 91 left cosets of H, (which is equivalent to the representation as a group of permutations of the roots) is primitive [13]. This proves Theorem 1. Unfor- tunately this criterion for indecomposability is not of much use. To prove Theorem 2 consider the lattice 9 of all fields L with k,(w) = k CL C K = k,(x) where w = f(x), and for each L let L* denote its com- pletion under the valuation corresponding to P(W). If L = kO(zl) where ?I is a polynomial in x then L* may be identified with the field k,(zl-‘) Gf formal power series in u-l. The mapping L ---fL", of the lattice 9 of fields between k and K into the lattice 59 of fields between K” and K* preserves containment and degree? because P(a) is totally ramified, so if Ll CL, then L,* CL,” and degree L,/L, = degree L,*/L, *. From this it follows that L* is isomorphic as a .&*-algebra to L ok R* so from properties of tensor products it follows that our mapling preserves intersections and composites. Hence, it is one-one. This much of the proof is valid for any field of constants k, . Assume now that k, is algebraically closed of characteristic 0. Then h* is isomorphic to a formal power series field k,(t) and is quasifinite, i.e., it has a cyclic extension of degree rz for every n and no other finite algebraic extensions. So in this case 9* is isomorphic to the lattice of subgroups of a cyclic group whose order is the degree of BY/k*. In this lattice deg(L,* n L,*)/k = g.c.d.(dsg L,*/~Y, deg L,“/ii*‘): it follows that these relations hold also in 9 aithough in general the fields in 8 may not even be normal over k. Moreover, in this case the fields in (7) are completely determined by the degrees of the polynomials .l” because the fields in 9* are completeiy determined by their degrees over k*. We finish the proof of Theorem 2 by the method used to prove the finite Jordan-Holder theorem: Let be maximal chains of fields. Assume r < s and use induction on P.