Another proof of Iwasawa's class number formula
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چکیده
منابع مشابه
Another Proof of the Harer-Zagier Formula
For a regular 2n-gon there are (2n − 1)!! ways to match and glue the 2n sides. The Harer-Zagier bivariate generating function enumerates the gluings by n and the genus g of the attendant surface and leads to a recurrence equation for the counts of gluings with parameters n and g. This formula was originally obtained using multidimensional Gaussian integrals. Soon after, Jackson and later Zagier...
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Let k be a number field and K/k a V4-extension, i.e., a normal extension with Gal(K/k) = V4, where V4 is Klein’s four-group. K/k has three intermediate fields, say k1, k2, and k3. We will use the symbol N i (resp. Ni) to denote the norm of K/ki (resp. ki/k), and by a widespread abuse of notation we will apply N i and Ni not only to numbers, but also to ideals and ideal classes. The unit groups ...
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We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform subl...
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متن کاملSpecial Cases of the Class Number Formula
Proof. Using that (Z/p)× is a cyclic group of order p − 1 (i.e. the existence of primitive roots), we see that there is a square root of −1 (that is, a non-trivial fourth root of 1) in (Z/p)× if and only if p ≡ 1 mod 4. Suppose now that p ≡ −1 mod 4, and suppose that α and β are two elements of Z[i] such that p|αβ. Then p = N(p)|N(α)N(β), and so (after relabelling if necessary) we may assume th...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1981
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-39-1-1-6