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On monogenity of certain pure number fields of degrees 2 r · 3 k · 7 s

Hamid Ben Yakkou, Jalal Didi (2024)

Mathematica Bohemica

Let K = ( α ) be a pure number field generated by a complex root α of a monic irreducible polynomial F ( x ) = x 2 r · 3 k · 7 s - m [ x ] , where r , k , s are three positive natural integers. The purpose of this paper is to study the monogenity of K . Our results are illustrated by some examples.

On octahedral extensions of and quadratic -curves

Julio Fernández (2003)

Journal de théorie des nombres de Bordeaux

We give a necessary condition for a surjective representation Gal ( ¯ / ) PGL 2 ( 𝔽 3 ) to arise from the 3 -torsion of a -curve. We pay a special attention to the case of quadratic -curves.

On power integral bases for certain pure number fields defined by x 18 - m

Lhoussain El Fadil (2022)

Commentationes Mathematicae Universitatis Carolinae

Let K = ( α ) be a number field generated by a complex root α of a monic irreducible polynomial f ( x ) = x 18 - m , m 1 , is a square free rational integer. 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.

On relative pure cyclic fields with power integral bases

Mohammed Sahmoudi, Mohammed Elhassani Charkani (2023)

Mathematica Bohemica

Let L = K ( α ) be an extension of a number field K , where α satisfies the monic irreducible polynomial P ( X ) = X p - β of prime degree belonging to 𝔬 K [ X ] ( 𝔬 K is the ring of integers of K ). The purpose of this paper is to study the monogenity of L over K by a simple and practical version of Dedekind’s criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field L with a...

On the compositum of all degree d extensions of a number field

Itamar Gal, Robert Grizzard (2014)

Journal de Théorie des Nombres de Bordeaux

We study the compositum k [ d ] of all degree d extensions of a number field k in a fixed algebraic closure. We show k [ d ] contains all subextensions of degree less than d if and only if d 4 . We prove that for d > 2 there is no bound c = c ( d ) on the degree of elements required to generate finite subextensions of k [ d ] / k . Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of d , but that one can take c = d when d is prime. This question was inspired by work of Bombieri and...

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