Relative integral basis for algebraic number fields

Abstract

At first conditions are given for existence of a relative integral basis for OK≅Okn−1⊕I with [K;k]=n. Then the constrtiction of the ideal I in OK≅Okn−1⊕I is given for proof of existence of a relative integral basis for OK4(m1,m2)/Ok(​m3). Finally existence and construction of the relative integral basis for OK6(n3,−3)/Ok3(n3),OK6(n3,−3)/Ok2(−3) for some values of n are given

A relative integral basis over ℚ(−3) for the normal closure of a pure cubic field

“…However, Haghighi's RIB for L/k contains two difficulties. The first is that in certain cases the RIB makes use of an element of norm 3 in a pure cubic field, a quantity which is not easy to determine, see [2,Theorem 5.1]. The second problem is that the RIB is not completely general, see [2,Theorem 5.3].…”