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Class Number One Problem for Imaginary Function Fields: The Cyclic Prime Power Case

✍ Scribed by Stéphan Sémirat

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
171 KB
Volume
84
Category
Article
ISSN
0022-314X

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✦ Synopsis

In this paper, we determine all finite separable imaginary extensions KÂF q (x) whose maximal order is a principal ideal domain in case KÂF q (x) is a non zero genus cyclic extension of prime power degree. There exist exactly 42 such extensions, among which 7 are non isomorphic over F q .

2000 Academic Press Theorem 0.1. Let l be a prime, and let n # N*. There are 42 cyclic imaginary l n -extensions KÂk such that the full constant field of K is F q , g K {0, and with ideal class number equal to one. They are defined by F q (x, y) with q, g K , defining equation and divisor class number h K as follows:

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