Working with your degree 8 polynomial over Q is almost certainly better. I would also recommend reducing the defining polynomial first:
sage: R.<g> = QQ[] sage: pol = g^8 - 5661818/709635*g^7 + 11951452814641/503581833225*g^6 - 5464287298588/167860611075*g^5 + 42311165180509/503581833225*g^4 + 290446480816/167860611075*g^3 + 6817133713732/503581833225*g^2 - 11294971392/55953537025*g + 2238425344/503581833225 sage: K.<a> = NumberField(pol) sage: K1=K.optimized_representation()[0]; K1 Number Field in a1 with defining polynomial x^8 - 2*x^7 - 2073127276349*x^6 - 585042438455127612*x^5 + 17251120619520968221641540*x^4 + 47323235466058260399591984538122*x^3 + 52569579991119152255555179191805210311*x^2 + 26979907667586120684167115024265757878264932*x + 5304889912416030130201287805372669997413025784321 -- not obviously a lot better, but at least it has integer coefficients. We can easily find its Galois group abstractly: sage: K1.galois_group(type='pari') Galois group PARI group [1152, -1, 47, "[S(4)^2]2"] of degree 8 of the Number Field in a1 with defining polynomial x^8 - 2*x^7 - 2073127276349*x^6 - 585042438455127612*x^5 + 17251120619520968221641540*x^4 + 47323235466058260399591984538122*x^3 + 52569579991119152255555179191805210311*x^2 + 26979907667586120684167115024265757878264932*x + 5304889912416030130201287805372669997413025784321 That code means its a double cover of S4^2 (1152 = 2*24^2). That's bigger than the 32 you were expecting though, so perhaps working with relative extensions would be better after all. John On 20 March 2018 at 15:44, Dima Pasechnik <[email protected]> wrote: > I do not know how efficiently relative extensions of Q[i] are implemented. > I would not mind an absolute field. > I need things like computing resultants and factoring univariate > polynomials to work not too slowly. > > I did some Groebner basis computation that gave me what I suspect is an > superfield of what I need, and it is of degree 32 over Q, with really huge > coefs.... > > -- > You received this message because you are subscribed to the Google Groups > "sage-nt" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at https://groups.google.com/group/sage-nt. > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "sage-nt" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/sage-nt. For more options, visit https://groups.google.com/d/optout.
