2016/07/11 by Zrinka Franušić, Franušić, Zrinka, Borka Jadrijević +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1607.03064
arxiv created 2016/07/11 · arxiv updated 2016/07/12
Let M be an imaginary quadratic field with the ring of integers ℤM and let ξ be a root of polynomial f( x) =x4-2cx3+2x2+2cx+1, where c∈ℤM, c∉\ 0,±2\. We consider an infinite family of octic fields Kc=M( ξ) with the ring of integers ℤ_Kc. Our goal is to determine all generators of relative power integral basis of O=ℤM[ ξ] over ℤM. 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 ℤM. We solve the system completely and find that all non-equivalent generators of power integral basis of O over ℤM are given by α=ξ, 2ξ-2cξ2+ξ3 for \vert c\vert ≥159108 and |c|≤200.