Function Repository Resource:

# GenericInvariant

Compute the generic invariant of a group

Contributed by: Tessa Wildsmith and Enrique Zeleny
 ResourceFunction["GenericInvariant"][gr] computes the generic invariant of a group gr of the same order as the group order. ResourceFunction["GenericInvariant"][gr,n] computes the generic invariant of a group gr of order n, greater than or equal to the group order.

## Details

Let G be a subgroup of the symmetric group Sn of permutations of n elements. A polynomial invariant of G is a polynomial expression IK [X1,x2,,Xn] which is left unchanged under a given set of algebraic transformations, namely the permutations of a given group G. In other words: for every σG, I(xσ(1)xσ(n)) = I(x1,,xn).
The computation of the Galois group of a polynomial requires a priori knowledge of the invariant functions that correspond to subgroups of Sn, hence reducing these invariants shall optimise the isolation of the Galois group. Let F*(x1,x2,,xn)=x1x2xn. Then define our function to be .
In the Stauduhar method, the invariant functions F that are used to isolate integer roots to the resolvent polynomial (a polynomial of lower degree whose roots are related to the roots of the original polynomial) of a given group.

## Examples

### Basic Examples (3)

Generic invariant of a permutation group:

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An equivalent specification:

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Using literals:

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### Scope (2)

Generic invariant of a symmetric group:

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The generic invariant can grow in size very quickly:

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### Properties and Relations (1)

Get the generic invariant of a polynomial using Stauduhar's method:

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Enrique Zeleny

## Version History

• 1.0.0 – 31 January 2022