On the discriminant of pure number fields

Let K = ℚ(α) be a number field generated by a complex root a of a monic irreducible polynomial ƒ (x) = x36 − m, with m ≠ ±1 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.

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On the construction of pure number fields of odd degrees with large $2$-class groups

Proceedings of the Japan Academy Series A Mathematical Sciences ◽

10.3792/pjaa.62.61 ◽

1986 ◽

Vol 62 (2) ◽

pp. 61-64

Author(s):

Shin Nakano

Keyword(s):

Number Fields ◽

Class Groups ◽

Pure Number

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On the Genus Fields of Pure Number Fields II

Tokyo Journal of Mathematics ◽

10.3836/tjm/1270215749 ◽

1981 ◽

Vol 04 (1) ◽

pp. 213-220

Author(s):

Makoto ISHIDA

Keyword(s):

Number Fields ◽

Pure Number

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On power integral bases for certain pure number fields defined by x24 – m

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2020.57.3.1472 ◽

2020 ◽

Vol 57 (3) ◽

pp. 397-407

Author(s):

Lhoussain El Fadil

Keyword(s):

Number Field ◽

Irreducible Polynomial ◽

Number Fields ◽

Rational Integer ◽

Complex Root ◽

Integral Bases ◽

Pure Number

AbstractLet 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.

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