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- W2034193857 abstract "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 factorizations (= 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 characteristic 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." @default.
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- W2034193857 date "1974-01-01" @default.
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- W2034193857 title "Prime and composite polynomials" @default.
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- W2034193857 doi "https://doi.org/10.1016/0021-8693(74)90023-4" @default.
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