An Infinite Family of Ninth Degree Dihedral polynomials

Let [Formula: see text] be an irreducible polynomial with rational coefficients, [Formula: see text] the number field defined by [Formula: see text], and [Formula: see text] the Galois group of [Formula: see text]. Let [Formula: see text], and let [Formula: see text] be the Galois group of [Formula: see text]. We investigate the extent to which knowledge of the conjugacy class of [Formula: see text] in [Formula: see text] determines the conjugacy class of [Formula: see text] in [Formula: see text]. We show that, in general, knowledge of [Formula: see text] does not automatically determine [Formula: see text], except when [Formula: see text] is isomorphic to [Formula: see text] (the cyclic group of order 4). In this case, we show [Formula: see text] is isomorphic to a non-split extension of [Formula: see text] (the dihedral group of order 8) by [Formula: see text]. We also show that [Formula: see text] is completely determined when [Formula: see text] is isomorphic to [Formula: see text] and [Formula: see text] is a perfect square. In this case, [Formula: see text].

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Galois groups of certain even octic polynomials

Chen

Chin

Tan

2022

J. Algebra Appl.

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Let [Formula: see text] be an irreducible polynomial where [Formula: see text] is a square. We give a method that completely describes the factorization patterns of a linear resolvent of [Formula: see text] using simple arithmetic conditions on [Formula: see text] and [Formula: see text]. As a result, we determine the exact six possible Galois groups of [Formula: see text] and completely classify all of them. As an application, we characterize the Galois groups of irreducible polynomials [Formula: see text]. We also use similar methods to obtain analogous results for the Galois groups of irreducible polynomials [Formula: see text].

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An Infinite Family of Ninth Degree Dihedral polynomials

Cited by 3 publications

References 7 publications

On the Galois group of a reciprocal even octic polynomial

On the Galois group of a reciprocal even octic polynomial

Galois groups of doubly even octic polynomials

Galois groups of certain even octic polynomials

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