engleski

Computing relative power integral bases in a family of quartic extensions of imaginary quadratic fields

Let M be an imaginary quadratic field with ring of integers Z_{; ; M}; ; . Let ξ be a root of the polynomial f(x)=x⁴-2cx³+2x²+2cx+1, c∈Z_{; ; M}; ; , c≠0. We consider an infinite family of octic fields K_{; ; c}; ; =M(ξ)with ring of integers Z_{; ; K_{; ; c}; ; }; ; . Since the integral basis of K_{; ; c}; ; is not known in a parametric form, our goal is to determine all generators of the O=Z_{; ; M}; ; [ξ] over Z_{; ; M}; ; (instead of Z_{; ; K_{; ; c}; ; }; ; over Z_{; ; M}; ; ). We show that our problem reduces to solving the system of relative Pellian equations cV²-(c+2)U²=-2μ, cZ²-(c-2)U²=2μ where μ is an unit in M. We solve the system completely and find that all non-equivalent generators of the power integral basis of O over Z_{; ; M}; ; are given by α=ξ, 2ξ-2cξ²+ξ³ for |c|≥159108.

relative power integral bases ; system of relative Pellian equations ; relative Thue equations

nije evidentirano