[GAP Forum] Choosing the right Field...

[Alexander Hulpke]

Dear GAP Forum,

Ravi Kulkarni write:
>  If I start with
> 
>  	R := PolynomialRing(Rationals,3);;
> 
>  I find that I should really have done
> 
>  	F := Field(Sqrt(3), E(4));;  	
> 	R := PolynomialRing(F,3);;
> 
>  instead, to get a nice decomposition in some example.
As far as creating polynomials is concernded, actually all polynomials created for the rationals, or any other subfield of the cyclotomic numbers live over the cyclotomics, i.e. you can just multiply them with cyclotomic numbers.

>   Is there some way to do this automatically without explicitly
> examining the eigenvalues/characters first?


I'm not sure what you mean by ``automatically''? If you just have polynomials you will have to investigate the roots. Are you asking for the smallest field of definition for all irreducible representations? (E.g. there is a theorem of Brauer stating that all character values will lie in Q(zeta) where zeta is an e-th root of unity for e the exponent of the finite group).

>  (And I think I now see why everybody tells me to work with
> permutation groups instead of matrix representations - I think those
> decompose over the Rationals only...)
Clearly not -- the regular representation e.g. contains all irreducibles.

Regards,

    Alexander Hulpke

-- Colorado State University, Department of Mathematics,
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