Kurt Foster on Sat, 03 May 2008 17:51:52 +0200

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Special Hilbert Class Field computation

I've got a bnf structure, field = bnfinit(poly) with field,clgp,no = 2. I want the Hilbert Class Field. A classical solution (due to Hilbert himself) is to find an integral element which is not a square, but is a square mod 4, i.e. mod 4zk. The bnf structure gives two lists of suspects: one is units mod squares of units. The other is a list of principal generators of P^2, where P is the ideal given by field.clgp.gen -- such a list can be produced using bnfisintnorm().

The thing that's got me hung up is checking for squareness mod 4. I can produce a list of squares mod 4 (the square roots are defined mod 2) and use nfbasistoalg() and nfalgtobasis() to get everything in terms of t_COL's of integer coefficients of basis elements to compare, but I'm probably being very clumsy about it. Besides, I'd probably be reinventing the wheel. Does anyone have a slick and quick way of doing this special computation?