Let be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields , which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).
We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate-Shafarevich groups of the associated elliptic curves. We also present a related criterion for primes p such that divides the class number of the imaginary quadratic field ℚ(√-p)....
We give a family of -polynomials with integer coefficients whose splitting fields over are unramified cyclic quintic extensions of quadratic fields. Our polynomials are constructed by using Fibonacci, Lucas numbers and units of certain cyclic quartic fields.
The aim of this paper is to prove the following Theorem Theorem Let be an octic subfield of the field and let be prime. Then divides if and only if divides for some , , , .