# On computing subfields. A detailed description of the algorithm

• Volume: 10, Issue: 2, page 243-271
• ISSN: 1246-7405

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## Abstract

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Let $ℚ\left(\alpha \right)$ be an algebraic number field given by the minimal polynomial $f$ of $\alpha$. We want to determine all subfields $ℚ\left(\beta \right)\subset ℚ\left(\alpha \right)$ of given degree. It is convenient to describe each subfield by a pair $\left(g,h\right)\in ℤ\left[t\right]×ℚ\left[t\right]$ such that $g$ is the minimal polynomial of $\beta =h\left(\alpha \right)$. There is a bijection between the block systems of the Galois group of $f$ and the subfields of $ℚ\left(\alpha \right)$. These block systems are computed using cyclic subgroups of the Galois group which we get from the Dedekind criterion. When a block system is known we compute the corresponding subfield using $p$-adic methods. We give a detailed description for all parts of the algorithm.

## How to cite

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Klüners, Jürgen. "On computing subfields. A detailed description of the algorithm." Journal de théorie des nombres de Bordeaux 10.2 (1998): 243-271. <http://eudml.org/doc/248167>.

@article{Klüners1998,
abstract = {Let $\mathbb \{Q\}\{(\alpha )\}$ be an algebraic number field given by the minimal polynomial $f$ of $\alpha$. We want to determine all subfields $\mathbb \{Q\}\{(\beta )\} \subset \mathbb \{Q\}\{(\alpha )\}$ of given degree. It is convenient to describe each subfield by a pair $(g , h) \in \mathbb \{Z\} [t] \times \mathbb \{Q\}[t]$ such that $g$ is the minimal polynomial of $\beta = h(\alpha )$. There is a bijection between the block systems of the Galois group of $f$ and the subfields of $\mathbb \{Q\}\{(\alpha )\}$. These block systems are computed using cyclic subgroups of the Galois group which we get from the Dedekind criterion. When a block system is known we compute the corresponding subfield using $p$-adic methods. We give a detailed description for all parts of the algorithm.},
author = {Klüners, Jürgen},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {subfields; block systems},
language = {eng},
number = {2},
pages = {243-271},
publisher = {Université Bordeaux I},
title = {On computing subfields. A detailed description of the algorithm},
url = {http://eudml.org/doc/248167},
volume = {10},
year = {1998},
}

TY - JOUR
AU - Klüners, Jürgen
TI - On computing subfields. A detailed description of the algorithm
JO - Journal de théorie des nombres de Bordeaux
PY - 1998
PB - Université Bordeaux I
VL - 10
IS - 2
SP - 243
EP - 271
AB - Let $\mathbb {Q}{(\alpha )}$ be an algebraic number field given by the minimal polynomial $f$ of $\alpha$. We want to determine all subfields $\mathbb {Q}{(\beta )} \subset \mathbb {Q}{(\alpha )}$ of given degree. It is convenient to describe each subfield by a pair $(g , h) \in \mathbb {Z} [t] \times \mathbb {Q}[t]$ such that $g$ is the minimal polynomial of $\beta = h(\alpha )$. There is a bijection between the block systems of the Galois group of $f$ and the subfields of $\mathbb {Q}{(\alpha )}$. These block systems are computed using cyclic subgroups of the Galois group which we get from the Dedekind criterion. When a block system is known we compute the corresponding subfield using $p$-adic methods. We give a detailed description for all parts of the algorithm.
LA - eng
KW - subfields; block systems
UR - http://eudml.org/doc/248167
ER -

## References

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