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

Autori
Franušić, Zrinka ; Jadrijević, Borka

Izvornik
Publicationes mathematicae (0033-3883) 92 (2018), 3-4; 293-315

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
index form equations, relative power integral basis, system of relative Pellian equations

Sažetak
Let M=Q(√(-D)) be an imaginary quadratic field with ring of integers Z_{; ; ; ; M}; ; ; ; and let ξ be a root of the polynomial f(x)=x⁴-2cx³+2x²+2cx+1, where c∈Z_{; ; ; ; M}; ; ; ; ∖{; ; ; ; 0, ±2}; ; ; ; and c≠±1 if D=1 or 3. We consider an infinite family of octic fields K_{; ; ; ; c}; ; ; ; =M(ξ) with the ring of integers Z_{; ; ; ; K_{; ; ; ; c}; ; ; ; }; ; ; ; . Our goal is to determine all generators of relative power integral basis of O=Z_{; ; ; ; M}; ; ; ; [ξ] 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 Z_{; ; ; ; M}; ; ; ; . We solve the system completely and find that all non-equivalent generators of power integral bases of O over Z_{; ; ; ; M}; ; ; ; are given by α=ξ, 2ξ-2cξ²+ξ³ for |c|≥159108 and |c|≤1000, c∉S_{; ; ; ; c}; ; ; ; (where S_{; ; ; ; c}; ; ; ; is a set of exceptional cases, |S_{; ; ; ; c}; ; ; ; |=28). Also, we find that, in all above cases, O admits no absolute power integral basis if -D≡2, 3(mod4).

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA