[GAP Forum] Understanding factorisation of polynomials using ALNUTH

[Bill Allombert]

On Wed, Apr 01, 2015 at 07:44:55AM +0530, Dr. Kashyap Rajeevsarathy wrote:
> Dear Forum,
> *gap> pol :=
> UnivariatePolynomial(Rationals,[6,0,-5,0,1]);x_1^4-5*x_1^2+6gap> L :=
> SplittingField(pol);<algebraic extension over the Rationals of degree
> 4>gap> FactorsPolynomialAlgExt(L,pol); [ x_1+(-1/20*a^3+9/10*a),
> x_1+(-1/40*a^3+19/20*a), x_1+(1/40*a^3-19/20*a), x_1+(1/20*a^3-9/10*a) ]*
> 
> How should we interpret the symbol 'a' here? 

a denotes any roots of L. You can do
a:=RootOfDefiningPolynomial(L);
to define a in GAP.

> In other words, is there is a
> way to realise the constant terms in these these symbolic expressions
> (involving a) in the factors as (plus or minus) Sqrt{2} and Sqrt{3}.

Just square them:
pol := UnivariatePolynomial(Rationals,[6,0,-5,0,1]);
L := SplittingField(pol);
FactorsPolynomialAlgExt(L,pol);
a:=RootOfDefiningPolynomial(L);

you get

gap> (-1/40*a^3+19/20*a)^2;
!3
gap> (-1/20*a^3+9/10*a)^2;
!2

> Maybe it is too much to ask, but in general, can GAP display these real
> roots as a list of radicals (for example Sqrt{2}, Sqrt{3} etc.), as it does
> while displaying characters.

I would suggest you define L explicitely as Q(sqrt(2),sqrt(3)), instead of using
SplittingField.

Cheers,
Bill.