On the ‘gap’ in a theorem of Heegner. (English) Zbl 0198.37702

Summary: In 1952, K. Heegner [Math. Z. 56, 227–253 (1952; Zbl 0049.16202)] gave a proof of the fact that there are exactly nine complex quadratic fields of class-number one. His proof rests on the fact that a certain 24th degree polynomial with rational coefficients has a 6th degree factor which also has rational coefficients. Unfortunately, this reducibility has never been justified. In this paper, we fill this gap in Heegner’s proof.


Zbl 0049.16202
Full Text: DOI


[1] Baker, A., Linear forms in the logarithms of algebraic numbers, Mathematika, 13, 204-216, (1966) · Zbl 0161.05201
[2] {\scBirch, B. J.} To appear.
[3] Dickson, L.E., ()
[4] Heegner, K., Diophantische analysis und modulfunktionen, Math Z., 56, 227-253, (1952) · Zbl 0049.16202
[5] Stark, H.M., A complete determination of the complex quadratic fields of class-number one, Mich. math. J., 14, 1-27, (1967) · Zbl 0148.27802
[6] Weber, H., ()
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