Let $K=\mathbb{Q}\left(\alpha \right)$ be a number field generated by a complex root $\alpha$ of a monic irreducible polynomial $f\left(x\right)=x^{18}-m$, $m\ne \mp 1$, is a square free rational integer. We prove that if $m\equiv 2$ or $3(\mathrm{mod}4)$ and $m\neg \equiv \mp 1(\mathrm{mod}9)$, then the number field $K$ is monogenic. If $m\equiv 1(\mathrm{mod}4)$ or $m\equiv 1(\mathrm{mod}9)$, then the number field $K$ is not monogenic.

We study the compositum $k^{\left[d\right]}$ of all degree $d$ extensions of a number field $k$ in a fixed algebraic closure. We show $k^{\left[d\right]}$ contains all subextensions of degree less than $d$ if and only if $d\le 4$. We prove that for $d\>2$ there is no bound $c=c\left(d\right)$ on the degree of elements required to generate finite subextensions of $k^{\left[d\right]}/k$. Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of $d$, but that one can take $c=d$ when $d$ is prime. This question was inspired by work of Bombieri and...

The main theorem gives necessary conditions and sufficient conditions for $\mathbf{Q}\left(\sqrt[9]{n}\right)$ to have class number prime to 3. These conditions involve only the rational prime factorization of $n$ and congruences mod 27 of the prime factors of $n$. They give necessary and sufficient conditions for most $n$.

We study the ramification properties of the extensions $\mathbb{Q}({\zeta }_m,\sqrt[m]{a})/\mathbb{Q}$ under the hypothesis that $m$ is odd and if $p\mid m$ than either $p\nmid v_p\left(a\right)$ or $p^{v_p\left(m\right)}\mid v_p\left(a\right)$ ($v_p\left(a\right)$ and $v_p\left(m\right)$ are the exponents with which $p$ divides $a$ and $m$). In particular we determine the higher ramification groups of the completed extensions and the Artin conductors of the characters of their Galois group. As an application, we give formulas for the $p$-adique valuation of the discriminant of the studied global extensions with $m=p^r$.

A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of $K^{\times }$ has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of $L^{\times }\setminus K^{\times }$. We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K.

We propose a definition of sign of imaginary quadratic fields. We give an example of such functions, and use it to define new invariants that are roots of the classical Ramachandra invariants. Also we introduce signed ordinary distributions and compute their signed cohomology by using Anderson's theory of double complex.