Pell-type equations and class number of the maximal real subfield of a cyclotomic field
- 124 Downloads
We investigate the solvability of the Diophantine equation \(x^2-my^2=\pm p\) in integers for certain integer m and prime p. Then we apply these results to produce family of maximal real subfield of a cyclotomic field whose class number is strictly larger than 1.
KeywordsDiophantine equation Real quadratic fields Maximal real subfield of cyclotomic field Class number
Mathematics Subject ClassificationPrimary: 11D09 11R29 Secondary: 11R11 11R18
The authors are indebted to the anonymous referee for his/her valuable suggestions which have helped improving the presentation of this manuscript.
- 2.Dickson, L.E.: History of the Theory of Numbers, vol. 2. Chelsea, New York (1952)Google Scholar
- 8.Sawilla, R.E., Silvester, A.K., Williams, H.C.: A New Look at an Old Equation. Algorithmic Number Theory (ANTS-VIII). Lecture Notes in Computer Science, vol. 5011. Springer, Berlin, pp. 37–59 (2008)Google Scholar
- 9.Serret, J.-A. (ed.): Oeuvres de Lagrange. I-XIV. Gauthiers-Villars, Paris (1877)Google Scholar