number fields - Subject to some conditions, is it possible to conclude a subfield of an abelian extension generated by a unit is a cyclic extension

My research is mostly in the area of modular categories. In the course of my research I came across a constraining set of number theoretic conditions that I'd like to exploit. It has been pointed out that several of the conditions seem a bit odd from a number theoretic point of view (which is perhaps why my attempts to find help in the literature have been fruitless), never-the-less they are what I have to work with. Discussing the source of the conditions would, I feel, take me to far afield from the question but if you're interested the major...Read more

"frequency" of fields for which the p-adic regulator vanishes (mod p)

There is a very nice question which arises in the study of theDiscrete Logarithm Problem which I wish to present here.The question, in a general setting, is to specify an empiricalexpression for the "frequency" of fields for which the $p$-adic regulatorvanishes $(\bmod p)$.More specifically: If $k$ is a number field and $p$ an odd prime, we let$$R_{p}(k) \in \operatorname{Rems} :=\{0, 1, ..., p-1\}$$be the remainder of the p-adic regulator modulo $p$.One would expect that the values of $R_{p}(k)$ are uniformly distributedwithin $\operatorname{R...Read more

number fields - A subring question (revised)

Hello, Let $K/{\mathbb Q}$ be a finite extension which is not necessarily Galois, and ${\mathcal O}$ be the ring of integers of $K$. Let $p$ be a prime in ${\mathbb Q}$ and let $p {\mathcal O}={\mathfrak p}_1^{k_1} \cdots {\mathfrak p}_r^{k_r}$ be its prime factorization in $K$. Next let ${\mathcal O}_i$ be the ring of integers of the completion of $K$ at ${\mathfrak p}_i$. Let ${\mathcal O}_p={\mathcal O}_1^{k_1} \times \cdots \times {\mathcal O}_r^{k_r}$. Is there a description of subrings $R$ of ${\mathcal O}_p$ such that $[{\mathcal O}_p:R ...Read more

number fields - Transformations of integer polynomials under combinations of their roots

I'm wondering whether the following ideas/questions give rise to an existing body of research. (Accordingly: please suggest appropriate changes to the tags!)PreambleWe consider polynomials f ∈ ℤ[x] with roots in ℝ, and for each polynomial f, the principal root is the real root with the largest magnitude. In the case of two roots of equal magnitude, we take the positive one.† So, for instance, √5 is the principal root of x2 − 5, the golden ratio φ is the principal root of x2 &minus...Read more

primitive elements - Do all algebraic number fields arise from Eisenstein polynomials?

This question came up while going through the application of Eisenstein criterion: The $p$-th cyclotomic polynomial after changing the variable $x$ to $(x+1)$ satisfies Eisenstein criterion. That is the minimal polynomial of $\zeta_p-1$ is an Eisenstein polynomial. Now let us take a general algebriac number field Q$[\alpha]=K$. Can one find another primitive element $\beta$ for $K$ such that its minimal polynomial is Eisenstein? As all quadratic fields arise from $\sqrt d$ which has equation $x^2-d$, it is true.Since the evidence so far is ...Read more