Polyadic rings, fields and integer numbers
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ID: 311716
2022
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Abstract
In chapter 4 polyadic rings and fields are introduced. Their direct products are unusual and can have different arity. The polyadic integer numbers, which form a polyadic ring, are defined. The basics of the new polyadic arithmetic thus introduced is presented: prime polyadic numbers, the polyadic Euler function, polyadic division with a remainder, etc. Secondary congruence classes of polyadic integer numbers and the corresponding finite polyadic rings are defined. A polyadic version of (prime) finite fields is introduced. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. None of the above situations is possible in the binary case. It is conjectured that a finite polyadic field should contain a certain canonical prime polyadic field, defined here as a minimal finite subfield, which can be considered as a polyadic analog ofGF(p). The Diophantine equations over the polyadic rings are then considered. Polyadic analogs of the Lander–Parkin–Selfridge conjecture and Fermat’s last theorem are formulated. For polyadic numbers neither of the standard statements holds. Polyadic versions of the Frolov’s theorem and the Tarry–Escott problem are presented.
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persistent_1771691201_6999dcc1602b6
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| Authors | Steven Duplij |
| Journal | science and technology of advanced materials |
| Year | 2022 |
| DOI |
10.1088/978-0-7503-2648-3ch4
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| URL | |
| Keywords | Keywords not found |
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