Does someone have a good and rigorous reference for the solution of quintic ploynomial equation with Jacobi Theta function, in English? Mathworld and Wikipedia don't give a good E …

Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$. Quest …

Hi, overflowers. I have a question concerning the torsion of elliptic curves over number fields. Let us consider an elliptic curve $E$ defined over ${\mathbb Q}$. From the Weil …

Let $K$ be a discrete valuation field with valuation $v:K\rightarrow \mathbb Z\cup {\infty}$ which is normalized by $v(\pi)=1$ for a prime element $\pi$. Let $v:\overline K\rightar …

A field is an ordered triple $(F, +,\cdot)$ of a set $F$ and binary operations $+,\times$ on $F$ such that $(F,+)$ and $(F\backslash 0,\times)$ are abelian groups satisfying the di …

Let $\mathbf C(t)$ be the field of rational functions and let $\overline{\mathbf C(t)}$ be an algebraic closure. Let $G$ be the Galois group of $\overline {\mathbf C(t)}$ over $\ma …

The definitions of a divisible group that I have seen all seem to assume abelian is an a priori property of the group. My question is as to whether or not it is known that--given a …

I have seen in many textbooks on analysis that the Archimedean property of reals is a consequence of the completeness axiom. However I am not convinced that we need to use such a p …

Let $P$ be a monic irreducible integral polynomial. Let $K=\mathbf Q[X]/(P)$ be the associated number field, $\mathcal O$ be its ring of integers and $R$ be the order $\mathbf Z[X] …

Let $x$ be $\cos \displaystyle \frac {2\pi} {n}$ for some natural number $n$. Then is there an integer $n$ such that $\mathbb{Q}(x^2+x)\neq \mathbb{Q}(x)$? I also would like to kno …

Observe that we have $Q(\sqrt{2})=Q((\sqrt{2}+1)^n)$. More generally, assume that $K$ is a finite extension of Q. Is there any $\alpha \in K$ such that $K=Q(\alpha^n)$ for every $ …

Lagrange proved that every (positive) rational integer is a sum of 4 squares. Are there general results like this for ring of integers of a number field? Is this class field theo …

Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$? I am going to be on the road soon, so pleas don't be offended if I don't respond qu …

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 h …

Dirichlet's unit theorem computes the group of units of the algebraic numbers of a number field. There are a few generalisations for orders available. Assume the order has an invo …