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According to page 1-2 of this paper (https://arxiv.org/abs/0906.4286), Mahler has established the inequality $$|\alpha_1 - \alpha_2| \geq H(\alpha_1)^{-(d-1)} \tag 1 $$ to be valid for all pairs of co …

I have been reading about the Schinzel-Zassenhauss conjecture, and have been looking for the following reference: Vinberg E.: On some number-theoretic conjectures of V. Arnold, Japanese J. Math., vol …

I have been reviewing some basic algebraic number theory and $p$-adic analysis, and the following thought crossed my mind: if $F/ \mathbb Q$ be a finite Galois extension, and $\eta$ be a $\mathbb Q$-a …

I have been trying to explore and learn about connections between the form class group and the ideal class group. To be on the same page, we define the form class group of a negative discriminant $D \ …

We know that since the ring $\mathbb Z [i]$ of Gaussian integers is a Principal Ideal Domain, the only integer primes which can norms of some ideal of $\mathbb Z [i]$ are those which can be expressed …

This is something that came out of working on a problem: Let $m$ be an odd positive integer and $f \in \mathbb Q[x]$ be a polynomial of degree less than $m$. With $\zeta_m$ denoting a primitive root …

This question is inspired from the post linked below: Can an algebraic number on the unit circle have a conjugate with absolute value different from 1? What I am curious about is the following: let $\ …

For the purposes of a project, I've been looking for the following two papers referred to in Serre's "Divisibilité de certaines fonctions arithmétiques": Landau (E.), - Über die Eitenlung der positi …

I came across the statement in a book: Let $k$ be a number field and $K$ be a Galois extension of $\mathbb Q$ containing $k$, with Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ and let $G_k:=\oper …

Consider a number field $K$, and let $v_1, \cdots v_n$ ($n \in \mathbb N$) be some finite (i.e. non-archimedean) places of $K$. Is the following true? For every $\alpha \in K^\times$ there exists $\b …

This is something that I have been thinking about for a while now, not sure if it is standard (or even true at all) or not: Let $K/ \mathbb Q$ be a CM number field, that is, it is closed under complex …

Let $a_1, \cdots, a_k \in [0, 1)$ be real numbers such that $1, a_1, \cdots, a_k$ are independent over the rational numbers. By the Weyl equidistribution criterion in $k$-dimensions, we know that the …

Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\sigm …

I have been reading section 7 of Serre's "Quelques applications du théorème de densité de Chebotarev" (http://www.numdam.org/item/PMIHES_1981__54__123_0/), and in particular have been trying to unders …

I have been studying about the Chebotarev Density Theorem and have been hunting for the following paper of Lagarias and Odlyzko for quite a while: Effective versions of the Chebotarev density theorem, …