Glasnik Matematicki, Vol. 40, No.1 (2005), 21-27.

A NOTE ON CLASS NUMBER ONE CRITERIA OF SIROLA FOR REAL QUADRATIC FIELDS

P. G. Walsh

Department of Mathematics, University of Ottawa, 585 King Edward St., Ottawa, Ontario, Canada K1N 6N5
e-mail: gwalsh@mathstat.uottawa.ca

Abstract.   In a recent paper, Širola gives two necessary and sufficient conditions for the class number of a real quadratic field to be equal to one. The purpose of this note is to remark that the equivalence of these conditions can be proved by using an elementary result of Nagell, which itself is a simple consequence of the fact that the Pell equation X2 - dY2 = 1 always has solutions in positive integers when d > 1 is squarefree.

2000 Mathematics Subject Classification.   11D09, 11R11, 11R29.

Key words and phrases.   Quadratic field, Pellian equation.

Full text (PDF) (free access)

DOI: 10.3336/gm.40.1.03

References:

  1. F. Lemmermeyer, Higher Descent on Conics. I. From Legendre to Selmer,
    arXiv: math.NT/0311309.

  2. R. A. Mollin, Quadratics, CRC Press, New York, 1995.

  3. R. A. Mollin, A continued fraction approach to the Diophantine equation ax2 - by2 = ± 1, JP Journal algebra, Number Theory and Appl. 4 (2004), 159-207.

  4. T. Nagell, On a special class of Diophantine equations of the second degree, Ark. Math. 3 (1954), 51-65.
    CrossRef

  5. K. Petr, On Pellian equation, Časopis Pest. Mat. Fys. 56 (1927), 57-66 (in Czech).

  6. B. Širola, Class number one quadratic fields and solvability of some Pellian equations, Acta. Math. Hungarica 104 (2004), 127-142.

  7. P. G. Walsh, The Pell Equation and Powerful Numbers, Master's Thesis, University of Calgary, 1988.
Glasnik Matematicki Home Page