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Hilberth’s Theorem 94 and Function Fields

In: Number Theory: New York Seminar 1991–1995

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  • Howard Kleiman

Abstract

Let f(x,y) be a monic absolutely irreducible polynomial in x of degree n with coefficients in Z[y]. If α is a root of f(x,y), L = Q(y)(α)/Q(y). Hilbert’s Theorem 94 [4] gives a procedure for determining rational primes p which divide the class number of a number field. Here an analogue of it is given for ordinary arithmetic function fields like L as defined by E. Weiss in [7]. A corollary of Theorem 1 is used to obtain rational prime divisors of class numbers of number fields L’ obtained from L by specialization of y into Z. Although the proof essentially follows that of Hilbert, use is made of the concept of NTU#x2019;s (non-trivial units) in fields like L. These units were implicity defined in [5].

Suggested Citation

  • Howard Kleiman, 1996. "Hilberth’s Theorem 94 and Function Fields," Springer Books, in: David V. Chudnovsky & Gregory V. Chudnovsky & Melvyn B. Nathanson (ed.), Number Theory: New York Seminar 1991–1995, chapter 17, pages 221-228, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-2418-1_17
    DOI: 10.1007/978-1-4612-2418-1_17
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