Function Repository Resource:

# GaloisGroupProperties

Compute the Galois group for a polynomial

Contributed by: Wolfram|Alpha Math Team
 ResourceFunction["GaloisGroupProperties"][poly,var] returns the Galois group for a univariate polynomial poly in the variable var. ResourceFunction["GaloisGroupProperties"][poly,var,prop] returns the specified property prop.

## Details and Options

ResourceFunction["GaloisGroupProperties"] typically supports only polynomials up to degree 4.
The property prop can be All, "Group", "GroupOrder", "Generators", "GroupElements", "MultiplicationTable" and "CayleyGraph". The default prop is "Group".

## Examples

### Basic Examples (1)

Compute the Galois group of the polynomial x2+1:

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### Scope (7)

Compute the Galois group of the polynomial x4+2:

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Return the CayleyGraph for the Galois group:

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Find the group order:

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Get the generators:

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Find the group elements:

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Display the group multiplication table:

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Get all of the available properties as an Association:

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### Properties and Relations (2)

An irreducible polynomial of prime degree p larger than 4 with exactly 2 nonreal roots has Galois group :

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Verify that there are 3 real roots:

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The Galois group for the irreducible polynomial of prime degree 5 with 2 nonreal roots is:

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## Publisher

Wolfram|Alpha Math Team

## Version History

• 2.0.0 – 23 March 2023
• 1.0.0 – 25 August 2020

## Author Notes

To view the full source code for GaloisGroupProperties, run the following code:

FileNameJoin[ReplacePart[FileNameSplit[FindFile["ResourceFunctionHelpers`"]], -1 "GaloisGroupProperties.wl"]] // SystemOpen