Editing Stark–Heegner theorem (section)

{{Short description|Quadratic imaginary number fields with unique factorisation}}
In [[number theory]], the '''Heegner theorem'''{{inconsistent|date=April 2025}}<ref>{{harvtxt|Elkies|1999}} calls this the Heegner theorem (cognate to Heegner points as in page xiii of {{harvtxt|Darmon|2004}}) but omitting Baker's name is atypical.{{inconsistent|date=April 2025}}{{harvtxt|Chowla|1970}} gratuitously adds Deuring and Siegel in his paper's title.</ref> establishes the complete list of the [[quadratic field|quadratic imaginary number fields]] whose [[ring of integers|rings of integers]] are  [[unique factorization domain|principal ideal domains.]]  It solves a special case of Gauss's [[Class number problem for imaginary quadratic fields|class number problem]] of determining the number of imaginary quadratic fields that have a given fixed [[ideal class group|class number]].

Let {{math|'''Q'''}} denote the set of [[rational number]]s, and let {{math|''d''}} be a [[square-free integer]]. The field {{math|'''Q'''({{radic|''d''}})}} is a [[quadratic extension]] of {{math|'''Q'''}}. The [[class number (number theory)|class number]] of {{math|'''Q'''({{radic|''d''}})}} is one [[if and only if]] the ring of integers of {{math|'''Q'''({{radic|''d''}})}} is a [[principal ideal domain]]. The Baker–Heegner–Stark theorem{{inconsistent|date=April 2025}} can then be stated as follows:

:If {{math|''d'' < 0}}, then the class number of {{math|'''Q'''({{radic|''d''}})}} is one if and only if <math>d \in \{\, -1, -2, -3, -7, -11, -19, -43, -67, -163\,\}.</math>
These are known as the [[Heegner number]]s.

By replacing {{mvar|d}} with the [[Discriminant of an algebraic number field|discriminant]] {{mvar|D}} of {{math|'''Q'''({{radic|''d''}})}} this list is often  written as:<ref>{{harvtxt|Elkies|1999}}, p.&nbsp;93.</ref>
:<math>D \in\{ -3, -4, -7, -8, -11, -19, -43, -67, -163\}.</math>