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Quadratic equations over p-adic fields and their applications in statistical mechanics

Mansoor Saburov^*, Mohd Ali Khameini Ahmad

 
ABSTRACT:     The p-adic models of statistical mechanics require the investigation of the roots of polynomial equations over p-adic fields in order to construct p-adic Gibbs measures. The most frequently asked question is that whether a root of a polynomial equation belongs to the domains ℤ_p^*, ℤ_p∖ℤ_p^*, ℤ_p, ℚ_p∖ℤ_p^*, ℚ_p∖(ℤ_p∖ℤ_p^*), ℚ_p∖ℤ_p, ℚ_p, 𝕊_p^m(0) or not. This question was open even for a quadratic equation. In this paper, by using the Newton polygon, we provide solvability criteria for quadratic equations over the domains mentioned above for all odd primes p. We also study the number of roots of quadratic equations over all domains given above. This study allows us to present a local description of roots of quadratic equations over p-adic fields whenever p>2.

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* Corresponding author, E-mail: msaburov@gmail.com

Received 16 Sep 2014, Accepted 8 Apr 2015