For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime ℘ of K, we determine the probability that ℘ splits into r primes in a random G-extension of K that is unramified at ℘. We find that these probabilities are nicely behaved and mostly ...

let d be a division ring with centre k and dim, d< ? a valuation on k and v a noninvariant extension of ? to d. we define the initial ramfication index of v over ?, ?(v/ ?) .let a be a valuation ring of o with maximal ideal m, and v , v ,…, v noninvariant extensions of w to d with valuation rings a , a ,…, a . if b= a , it is shown that the following conditions are equivalent: (i) b is a finite...

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let K− denote the maximal abelian p-extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the gene...

Let p be an odd prime number, k an imaginary abelian field containing a primitive p-th root of unity, and k∞/k the cyclotomic Zp-extension. Denote by L/k∞ the maximal unramified pro–p abelian extension, and by L′ the maximal intermediate field of L/k∞ in which all prime divisors of k∞ over p split completely. Let N/k∞ (resp. N ′/k∞) be the pro–p abelian extension generated by all p-power roots ...

Let p and q be two positive primes, let \(\ell\) an odd prime F a quadratic number field. K extension of degree such that is dihedral \({\mathbb {Q}}\), or else abelian \(\ell\)-extension unramified over whenever divides the class F. In this paper, we provide complete characterization division quaternion algebras \(H_{K}(p, q)\) K.

Let k be a real quadratic field and p an odd prime number which splits in k. In a previous work, the author gave a sufficient condition for the Iwasawa invariant λp(k) of the cyclotomic Zp-extension of k to be zero. The purpose of this paper is to study the case p = 3 of this result and give new examples of k with λ3(k) = 0, by using information on the initial layer of the cyclotomic Z3-extensi...

Let L = K() be an abelian extension of degree n of a number eld K, given by the minimal polynomial of over K. We describe an algorithm for computing the local Artin map associated to the extension L=K at a nite or innnite prime v of K. We apply this algorithm to decide if a nonzero a 2 K is a norm from L, assuming that L=K is cyclic.

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...

Let F be a field of characteristic 2 and let K/F be a purely inseparable extension of exponent 1. We show that the extension is excellent for quadratic forms. Using the excellence we recover and extend results by Aravire and Laghribi who computed generators for the kernel Wq(K/F ) of the natural restriction map Wq(F ) → Wq(K) between the Witt groups of quadratic forms of F and K, respectively, ...

Let k be a number field and K a finite extension of k. We count points of bounded height in projective space over the field K generating the extension K/k. As the height gets large we derive asymptotic estimates with a particularly good error term respecting the extension K/k. In a future paper we will use these results to get asymptotic estimates for the number of points of fixed degree over k...