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Split Primes and Integer-Valued Polynomials

✍ Scribed by D. Mcquillan

Publisher: Elsevier Science
Year: 1993
Tongue: English
Weight: 159 KB
Volume: 43
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.1993.1019

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✦ Synopsis

Let (R) be a Dedekind domain with field of fractions (K, L=K(x)) a finite separable extension of (K), and (S) the integral closure of (R) in (L). Let (I) be the subring of (K[X]) consisting of all polynomials (g(x)) in (K[X]) such that (g(R) \subset R), and let (E_{x}: I \rightarrow L) be the evaluation map defined by (E_{x}(g(x))=g(x)). Then (E_{\mathrm{x}}(l)) is precisely the overring of (S) determined by the prime ideals (P) of (S) which are split completely over (R) and at which (x) is integral. This answers a question posed by R. Gilmer and W. W. Smith (1985, Houston J. Math. 11, No. 1, 65 74) in connection with the ideal structure of (I) and solved by them when (R=\mathbf{Z}) and (L=\mathbf{Q}(\sqrt{d})). . 1993 Actademic Press, inc.

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