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Primitive element for a number field, and ramification

Let $K=\mathbb Q(\theta)$ be a number field with integral primitive element $\theta$, and let $f(x)$ be the minimal polynomial of $\theta$. Let $p$ be a rational prime. It's well known that if $p$ ...
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Divisor bounds of ideals in number fields

Let $K$ be an algebraic number field and let $I$ be an ideal in $O_K$ (the ring of integers). Denote by $d(I)$ the number of ideals that divide $I$. So if $I= \prod_{i=1}^k p_i^{e_i}$ is the ...
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Simultaneous Powers Far From 1

I'm looking for a reference or proof of the following. Let $K/\mathbb{Q}$ be a finite Galois extension of degree $n$. Let $a_1,\ldots,a_n$ be Galois conjugate elements in the ring of integers of $K$ ...
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Irreducibility of the trinomial over Q

I'm trying to find an algebraic proof of irreducibility of the polynomial $x^n-x-1$ over rational numbers (or integers, which the same). I've read the Selmer's paper "On the irreducibility of certain ...
Fix an algebraic integer $\alpha$ of degree $n$ such that the extension $K=\mathbf{Q}(\alpha)/\mathbf{Q}$ has intermediate fields. (We can assume $K$ is Galois with non-simple Galois group.) This $\... 2answers 1k views Quintic polynomial solution by Jacobi Theta function. Does someone have a good and rigorous reference for the solution of quintic ploynomial equation with Jacobi Theta function, in English? Mathworld and Wikipedia don't give a good English reference, at ... 0answers 151 views Full$n$-torsion of elliptic curves and the cyclotomic field of order$n$Hi, overflowers. I have a question concerning the torsion of elliptic curves over number fields. Let us consider an elliptic curve$E$defined over${\mathbb Q}$. From the Weil pairing one can ... 0answers 1k views Orders in number fields Let$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$. Question: Let$p$be an ... 1answer 153 views ramification of discrete valuation field Let$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\rightarrow \mathbb Q\cup\{...
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 distributive laws \$\...