ON THE REFLECTION THEOREM IN PRIME CYCLOTOMIC FIELDS
نویسندگان
چکیده
منابع مشابه
Class numbers of real cyclotomic fields of prime conductor
The class numbers h+l of the real cyclotomic fields Q(ζl + ζ −1 l ) are notoriously hard to compute. Indeed, the number h+l is not known for a single prime l ≥ 71. In this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic fields Q(ζl + ζ −1 l ) for the primes l < 10, 000. It is quite likely that these subgroups are in fact equal to the class ...
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Proof. (i) We have (1− ζ pn)/(1− ζpn) = 1+ ζpn + · · ·+ ζ i−1 p ∈ Z[ζpn ]. On the other hand, if ii ≡ 1 (mod p), we have (1 − ζpn)/(1 − ζ i pn) = (1 − ζ ii′ pn)/(1 − ζ i pn) = 1 + ζ p + · · ·+ ζ i(i′−1) p ∈ Z[ζpn ], so we find that (1 − ζ i pn) and (1− ζpn) divide one another in Z[ζpn] and hence in OQ(ζpn). (ii) By (i), 1 + ζpn = (1− ζ 2 pn)/(1− ζpn) is a unit in Z[ζpn ], hence in OQ(ζpn). (iii...
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For an imaginary quadratic number field K and an odd prime number l, the anti-cyclotomic Zl-extension of K is defined. For primes p of K, decomposition laws for p in the anti-cyclotomic extension are given. We show how these laws can be applied to determine if the Hilbert class field (or part of it) of K is Zl-embeddable. For some K and l, we find explicit polynomials whose roots generate the f...
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ژورنال
عنوان ژورنال: Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics
سال: 1972
ISSN: 1883-2172,0373-6385
DOI: 10.2206/kyushumfs.26.333