Semantic Apparatus – Shapes of Pure Prime Degree Number Fields

Cited by Lee Sonogan

A Big Question About Prime Numbers Gets a Partial Answer | WIRED

Abstract by Holmes, Erik

This thesis extends a result of Rob Harron, in [Har17]. Specifically, Harron studies the shapes of pure cubic number fields K = Q( 3 √m) and shows that the shape is a complete invariant of the family of pure cubic number fields, and that the shapes are equidistributed on one-dimensional subspaces of the space of shapes. For pure prime degree number fields, K = Q( p √m), we show that the shape is a complete invariant. For  = p−1 2 , our main result shows that the shapes of these fields lie on one of two -dimensional subspaces of the space of shapes and we prove equidistribution results for p < 1000. This work uses analytic methods which differ from those used in [Har17] and we therefore obtain an alternative proof of his result as well. We also prove that the family of pure prime degree number fields is equivalently the family of degree p number field with Galois group Fp and fixed resolvent field Q(ζp). This allows us to rephrase our results in a manner more closely related to those in the study of number field asymptotics and specifically Malle’s conjecture. This alternative also allows us to ask a very natural follow up question which we intend to investigate in future work.

Publication: University of Hawai’i at Manoa, ProQuest Dissertations Publishing (Peer-Reviewed Journal)

Pub Date: Aug 2021 Doi:

Keywords: Semantic Shapes, Pure Prime, Degree Number Field (Plenty more sections and references in this research article)

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