Let m ≠ 0, ±1 and n ≥ 2 be integers. The ring of algebraic integers of the pure fields of type is explicitly known for n = 2, 3,4. It is well known that for n = 2, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of m modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases.
In this paper we explicitly give an integral basis of the field , where m ≠ ±1 is square-free. Furthermore, we show that similarly to the quadratic case, an integral basis of is repeating periodically in m with period length depending on n.
Let K = ℚ(α) be a number field generated by a complex root α of a monic irreducible polynomial f(x) = x24 – m, with m ≠ 1 is a square free rational integer. In this paper, we prove that if m ≡ 2 or 3 (mod 4) and m ≢∓1 (mod 9), then the number field K is monogenic. If m ≡ 1 (mod 4) or m ≡ 1 (mod 9), then the number field K is not monogenic.
be an algebraic number which is not a root of a rational number. We show that the discriminant of
tends to infinity with
tending to infinity and give a lower bound for this discriminant in terms of the degree of
, its Mahler’s measure and