DISCRIMINANT and integral basis OF ℚ(√[12]a)

2023/04/01 by Jakhar, Anuj, Kumar, Surender
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2304.00339

Abstract

Suppose m be a 12-th power free integer. Let K=ℚ(θ) be an algebraic number field defined by a complex root θ of an irreducible polynomial x¹²-m and O_K be its ring of integers. In this paper, we determine the highest power of p dividing the index of the subgroup \Z[θ] in O_K and p-integral basis of K for each prime p. These p-integral bases lead to the construction of an integral basis of K which is illustrated with examples. In particular, when m is a square free integer, we provide necessary and sufficient conditions for the set \1,θ,θ²,⋯,θ¹⁰,θ¹¹\ to be an integral basis of K.

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