On index divisors and monogenity of certain octic number fields defined by x8+ax3+b
DOI:
https://doi.org/10.3842/umzh.v76i7.7536Keywords:
Theorem of Dedekind, Theorem of Ore, prime ideal factorization, Newton polygon, Index of a number field, Power integral basis, MonogenicAbstract
UDC 511
For any octic number field K generated by a root α of a monic irreducible trinomial F(x)=x8+ax3+b∈Z[x] and for every rational prime p, we show when p divides the index of K. We also describe the prime power decomposition of the index i(K). In this way, we give a partial answer to { Problem 22} of Narkiewicz [Elementary and analytic theory of algebraic numbers, Springer-Verlag, Auflage (2004)] for this family of number fields. As an application of our results, we conclude that if i(K)≠1, then K is not monogenic. We illustrate our results by some computational examples.
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