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18 votes
4 answers
1k views

In which cyclic cubic number fields does there exist this type of unit?

Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $\mathcal{O}_K$. Define $K$ to be blue if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/...
Christine McMeekin's user avatar
9 votes
1 answer
737 views

Square root in number field

I'm trying implement an algorithm that, for an element $b$ of a number field $\mathbb{Q}(\alpha)$, if it is a square in $\mathbb{Q}(\alpha)$ (i.e., $\exists x\in\mathbb{Q}(\alpha):x^2=b$), computes ...
Tippisum's user avatar
  • 153
7 votes
1 answer
623 views

Algorithm for computing whether a cubic field is monogenic?

I am interested in existing algorithms to compute whether a given non-cyclic, non-pure cubic extension $K/\mathbb{Q}$ is monogenic or not, and if so, to give me a defining polynomial for the integral ...
edward cornfoot's user avatar
5 votes
0 answers
741 views

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 $p$ ...
352506's user avatar
  • 1,021
2 votes
1 answer
157 views

$f(x)\bmod p$ and decomposition of prime ideals

While reading Serre's beautiful book Lectures on $N_X(p)$, I thought of a related question. Let $f(x)\in \mathbb{Z}[x]$ be a monic irreducible polynomial with integer coefficients. Let $K$ be the ...
youknowwho's user avatar
0 votes
0 answers
149 views

How to determine if a unramifed prime split or not?

Let $K$ be the Number field and $L$ be finite extension where $\mathfrak{p}$ prime of K is unramified. Are there any conditions on $\mathfrak{p}$ so that I can say $\mathfrak{p}$ splits completely in ...
SUNIL PASUPULATI's user avatar