# The Complexity of Computing all Subfields of an Algebraic Number Field

@article{Szutkoski2019TheCO,
title={The Complexity of Computing all Subfields of an Algebraic Number Field},
author={Jonas Szutkoski and M. V. Hoeij},
journal={J. Symb. Comput.},
year={2019},
volume={93},
pages={161-182}
}
• Published 2019
• Mathematics, Computer Science
• J. Symb. Comput.
For a finite separable field extension K/k, all subfields can be obtained by intersecting so-called principal subfields of K/k. In this work we present a way to quickly compute these intersections. If the number of subfields is high, then this leads to faster run times and an improved complexity.

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#### References

SHOWING 1-10 OF 34 REFERENCES
Generating subfields
• Mathematics, Computer Science
• ISSAC '11
• 2011
The notion of generating subfields is introduced, a set of up to n subfields whose intersections give the rest, and an efficient algorithm which uses linear algebra in k or lattice reduction along with factorization is provided. Expand
Computing subfields: Reverse of the primitive element problem
• Mathematics
• 1993
We describe an algorithm which computes all subfields of an effectively given finite algebraic extension. Although the base field can be arbitrary, we focus our attention on the rationals.
Algebraic factoring and rational function integration
• B. Trager
• Mathematics, Computer Science
• SYMSAC '76
• 1976
A new, simple, and efficient algorithm for factoring polynomials in several variables over an algebraic number field is presented and a constructive procedure is given for finding the least degree extension field in which the integral can be expressed. Expand
Block Systems of a Galois Group
• A. Hulpke
• Mathematics, Computer Science
• Exp. Math.
• 1995
An algorithm to compute subfields of an algebraic number field as block systems of its Galois group as well as symbolic computations and avoids numerical approximations is described. Expand
On computing subfields. A detailed description of the algorithm
Let Q(03B1) be an algebraic number field given by the minimal polynomial f of a. We want to determine all subfields Q(03B2) ~ Q(03B1) of given degree. It is convenient to describe each subfield by aExpand
Sharp estimates for triangular sets
• Mathematics, Computer Science
• ISSAC '04
• 2004
Polynomial bounds are proved in terms of intrinsic quantities for the height and degree of the coefficients of triangular sets of zero-dimensional varieties defined over the rational field. Expand
Powers of tensors and fast matrix multiplication
• F. Gall
• Mathematics, Computer Science
• ISSAC
• 2014
This paper presents a method to analyze the powers of a given trilinear form and obtain upper bounds on the asymptotic complexity of matrix multiplication and obtains the upper bound ω < 2.3728639 on the exponent of square matrix multiplication, which slightly improves the best known upper bound. Expand
A relative van Hoeij algorithm over number fields
• K. Belabas
• Computer Science, Mathematics
• J. Symb. Comput.
• 2004
Abstract van Hoeij’s algorithm for factoring univariate polynomials over the rational integers rests on the same principle as the Berlekamp–Zassenhaus algorithm, but uses lattice basis reduction toExpand
Isomorphisms of Algebraic Number Fields
• Mathematics, Computer Science
• ArXiv
• 2010
A new method to find (if they exist) all isomorphisms, $\mathbb{Q}(\beta) \rightarrow \mathbb(Q)(\alpha)$, which is particularly efficient if the number of isomorphism is one. Expand
Some Useful Bounds
Some fundamental inequalities for the following values are listed: the determinant of a matrix, the absolute value of the roots of a polynomial, the coefficients of divisors of polynomials, and theExpand