Karim BELABAS on Sun, 15 Jun 2003 14:25:00 +0200 (MEST) |
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Re: Finding an integral basis of a relative extension of nf's |
On Sun, 15 Jun 2003, Bill Hale wrote: > At 10:23 PM -0400 6/14/03, Mak Trifkovic wrote: >>I have a number field K=bnfinit(P(y)), and an extension L given by a >>polynomial Q(x) in K[x]. Assuming K has class number one,how do I get >>gp to find an integral basis for O_L over O_K, expressed as a set of >>polynomials in x with coefficients in Q[y]? > I am new to Pari myself. I am not sure if the following is > totally correct. Look at the following and refer to the > manual for details. You may wish to try your own polynomials > to test. > > ================================== > py=y^2+y+1 > qx=x^6+x^5+x^4+x^3+x^2+x+1 > K=bnfinit(py) > L=rnfinit(K,qx) > K.zk > L[7][1] > lift(L[7][1]) The above is incorrect in general: it assumes the ideals in L[7][2] are all equal to O_K, i.e represented by identity matrices (which is indeed true in thsi specific case. [ Background: L[7] is only a pseudo-basis (see rnfpseudobasis()) for O_L, not a true O_K-basis, since the latter does not exist in general. And can be much more expensive to compute even if it does exist ]. What you're looking for is rnfbasis(bnfinit(py), qx) Cheers, Karim. -- Karim Belabas Tel: (+33) (0)1 69 15 57 48 Dép. de Mathématiques, Bât. 425 Fax: (+33) (0)1 69 15 60 19 Université Paris-Sud http://www.math.u-psud.fr/~belabas/ F-91405 Orsay (France) http://www.parigp-home.de/ [PARI/GP]