Heegner's lemma

Short description: Criterion for existence of polynomial roots

In mathematics, Heegner's lemma is a lemma used by Kurt Heegner in his paper on the class number problem. His lemma states that if

$y^2=a_4{x}^4+a_3{x}^3+a_2{x}^2+a_{1}x+a_0$

is a curve over a field with a₄ not a square, then it has a solution if it has a solution in an extension of odd degree.

References

• Birch, Bryan (2004), "Heegner points: the beginnings", in Darmon, Henri; Zhang, Shou-Wu, Heegner points and Rankin L-series, Math. Sci. Res. Inst. Publ., 49, Cambridge University Press, pp. 1–10, doi:10.1017/CBO9780511756375.002, ISBN 978-0-521-83659-3