Journal of East China Normal University(Natural Sc ›› 2009, Vol. 2009 ›› Issue (5): 138-141.

• Article • Previous Articles    

Solutions to some Diophantine equations over mathbf{Q}(\sqrt{-3}

WANG Yong-liang   

  1. Department of Mathematics, Heze University, Heze, 274015, China
  • Received:2008-09-24 Revised:2009-02-10 Online:2009-09-25 Published:2014-10-13
  • Contact: WANG Yong-liang

Abstract: By using Fermat’s method of descent, this paper proved that Diophantine equations { x^{4}-y^{4}=z^{2}} and { x^{4}+4y^{4}=z^{2}} have no non-trivial solutions over {\mathbf{Q}}(\sqrt{-3}), which implies that the Fermat Equation also has no non-trivial solutions in this field for n =4.

Key words: \quad ring of algebraic integers, \quad imaginary quadratic fields, Fermat’s method of descent, \quad ring of algebraic integers, \quad imaginary quadratic fields

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