Browsing by Author "Mrema, Elizabeth"

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    Study of cyclotomic extensions of degree power of 2 and classification of radical extensions up to isomorphism
    (Stellenbosch : Stellenbosch University, 2024-03) Mrema, Elizabeth; Marques, Sophie; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    In this thesis, we gain a deeper understanding of cyclotomic extensions of degree powers of 2 and the classification of radical extensions (both separable and insepa‑ rable) up to isomorphism. Our main results about cyclotomic extensions of degree power of 2 describe their Galois structures, their degrees, their subextensions, their tower decompositions, and the minimal polynomials of some traces of root of unity generating all their subsextensions over an arbitrary base field. In exploring these as‑ pects, we discover two important invariants 𝓁𝑝∞ and 𝜈𝑝∞ where 𝑝 is a prime number, holding essential information about cyclotomic extensions of degree 2 and those gen‑ erated by primitive (2𝑒)𝑡ℎ roots of unity where 𝑒 ∈ ℕ. In our quest to provide explicit expressions for the coefficients of the minimal polynomials of the subextensions of cyclotomic extensions generated by primitive (2𝑒)𝑡ℎ root of unity, we discover fasci‑ nating characterizations, some of which are linked to the well‑known Catalan num‑ bers solving Combinatorial problems using field theory. Building upon the insights gained from our exploration of cyclotomic extensions, we provide a comprehensive classification of separable and inseparable radical ex‑ tensions up to isomorphism. In order to have a global understanding of these exten‑ sions up to isomorphism, we exhibit a meaningful parameterization of the set of iso‑ morphic radical extensions into moduli spaces involving the action of some groups.