منابع مشابه
Kuroda’s Class Number Formula
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 ...
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...
متن کاملOn a Class Number Formula for Real Quadratic Number Fields
For an even Dirichlet character , we obtain a formula for L(1;) in terms of a sum of Dirichlet L-series evaluated at s = 2 and s = 3 and a rapidly convergent numerical series involving the central binomial coeecients. We then derive a class number formula for real quadratic number elds by taking L(s;) to be the quadratic L-series associated with these elds.
متن کاملLecture # 8 and 9 : Ideals and the Class Number Formula
There’s a different way to go about the problem of finding all ways of writing n = x + y. Recall that in the Gaussian integers Z[i] we have an automorphism α 7→ ᾱ. Thus the function Nα = αᾱ = x + y is multiplicative. Thus our question is just to find all ways of writing n as a norm from the Gaussian integers. To answer this question we need to know a little about the structure of Z[i]. (Note th...
متن کاملThe Dirichlet Class Number Formula for Imaginary Quadratic Fields
because 2, 3, and 1± √ −5 are irreducible and nonassociate. These notes present a formula that in some sense measures the extent to which unique factorization fails in environments such as Z[ √ −5]. Algebra lets us define a group that measures the failure, geometry shows that the group is finite, and analysis yields the formula for its order. To move forward through the main storyline without b...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1994
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-66-3-245-260