Class groups of characteristic-p function field analogues of Q(n^(1/p))
| dc.contributor.advisor | Washington, Lawrence | en_US |
| dc.contributor.author | Reich, Steven | en_US |
| dc.contributor.department | Mathematics | en_US |
| dc.contributor.publisher | Digital Repository at the University of Maryland | en_US |
| dc.contributor.publisher | University of Maryland (College Park, Md.) | en_US |
| dc.date.accessioned | 2021-07-14T05:31:37Z | |
| dc.date.available | 2021-07-14T05:31:37Z | |
| dc.date.issued | 2021 | en_US |
| dc.description.abstract | In the theory of cyclotomic function fields, the Carlitz module $\Lambda_M$ associated to a polynomial $M$ in a global function field of characteristic $p$ provides a strong analogy to the roots of unity $\mu_p$ in a number field. In this work, we consider a natural extension of this theory to give a compatible analogue of the $p$-th root of an integer $n$. The most fundamental case, and the one which most closely mimics the number field situation, is when the Carlitz module is defined by a linear polynomial (which can be assumed to be $T$) in $k={\mathbb F}_q(T)$. The Carlitz module $\Lambda_T$ generates a degree-$(q-1)$ extension $k(\Lambda_T)$ which shares many properties with the field ${\mathbb Q}(\mu_p)$, where $\mu_p$ is the module of $p$-th roots of unity. To form the analogue of ${\mathbb Q}(\sqrt[p]{n})$, we define a degree-$q$ extension $F/k$ associated to a polynomial $P(T) \in k$, for which the normal closure is formed by adjoining $\Lambda_T$. In the introduction, we describe in detail the parallels between this construction and that in the number field setting. We then compute the class number $h_F$ for a large number of such fields. The remainder of the work is concerned with proving results about the class groups and class numbers of this family of fields. These are:\begin{itemize} \item a formula relating the class number of $F$ to that of its normal closure, along with a theorem about the structure of the class group of the normal closure \item a formula relating the class number of a compositum of such $F$ to the class numbers of the constituent fields \item conditions on $P(T)$ for when the characteristic, $p$, of $F$ divides its class number, along with bounds on the rank of the $p$-part of the class group. \end{itemize} | en_US |
| dc.identifier | https://doi.org/10.13016/uaqm-iauw | |
| dc.identifier.uri | http://hdl.handle.net/1903/27427 | |
| dc.language.iso | en | en_US |
| dc.subject.pqcontrolled | Mathematics | en_US |
| dc.subject.pqcontrolled | Theoretical mathematics | en_US |
| dc.subject.pquncontrolled | Artin-Schreier extension | en_US |
| dc.subject.pquncontrolled | Carlitz module | en_US |
| dc.subject.pquncontrolled | Class group | en_US |
| dc.subject.pquncontrolled | Cyclotomic field | en_US |
| dc.subject.pquncontrolled | Function field | en_US |
| dc.subject.pquncontrolled | Galois cohomology | en_US |
| dc.title | Class groups of characteristic-p function field analogues of Q(n^(1/p)) | en_US |
| dc.type | Dissertation | en_US |
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