A natural graph of finite fields distinguishing between models |
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Institution: | 1. Section de mathématiques, Université de Genève, 2-4 Rue du Lièvre, Case Postale 64, 1211 Genève 4, Switzerland;2. Matematiska institutionen, Uppsala universitet, Box 256, 751 05 Uppsala, Sweden |
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Abstract: | We define a graph structure associated in a natural way to finite fields that nevertheless distinguishes between different models of isomorphic fields. Certain basic notions in finite field theory have interpretations in terms of standard graph properties. We show that the graphs are connected and provide an estimate of their diameter. An accidental graph isomorphism is uncovered and proved. The smallest non-trivial Laplace eigenvalue is given some attention, in particular for a specific family of 8-regular graphs showing that it is not an expander. We introduce a regular covering graph and show that it is connected if and only if the root is primitive. |
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Keywords: | Graphs of finite fields Models of finite fields |
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