Bill Hale on Sun, 15 Jun 2003 07:02:04 -0500 |
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Re: Finding an integral basis of a relative extension of nf's |
At 10:23 PM -0400 6/14/03, Mak Trifkovic wrote: >Hi, > >I need help with the following: > >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]? > >thanks, >Mak 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 output that I get for last three statements is: ? %5 = [1, y] ? %6 = [Mod(1, y^2 + y + 1), Mod(1, y^2 + y + 1)*x, Mod(1, y^2 + y + 1)*x^2, Mod(1, y^2 + y + 1)*x^3, Mod(1, y^2 + y + 1)*x^4, Mod(1, y^2 + y + 1)*x^5] ? %7 = [1, x, x^2, x^3, x^4, x^5] -- Bill Hale