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Stark–Heegner theorem
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<p>In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, the Heegner theorem[<i>inconsistent</i>] establishes the complete list of the <a href="/facts/Quadratic_field/dUnSDN4e">quadratic imaginary number fields</a> whose <a href="/facts/Ring_of_integers/KEzxxMbh">rings of integers</a> are <a href="/facts/Unique_factorization_domain/aYsX2swC">principal ideal domains.</a> It solves a special case of Gauss's <a href="/facts/Class_number_problem_for_imaginary_quadratic_fields/bIrLDjde">class number problem</a> of determining the number of imaginary quadratic fields that have a given fixed <a href="/facts/Ideal_class_group/LLn5Pwj1">class number</a>. </p><p>Let Q denote the set of <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>, and let <i>d</i> be a <a href="/facts/Square-free_integer/R3qR9hP1">square-free integer</a>. The field Q(√<i>d</i>) is a <a href="/facts/Quadratic_extension/DWBtcL4A">quadratic extension</a> of Q. The <a href="/facts/Class_number_(number_theory)/LLn5Pwj1">class number</a> of Q(√<i>d</i>) is one <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the ring of integers of Q(√<i>d</i>) is a <a href="/facts/Principal_ideal_domain/JCwAUnyW">principal ideal domain</a>. The Baker–Heegner–Stark theorem[<i>inconsistent</i>] can then be stated as follows: </p> If <i>d</i> < 0, then the class number of Q(√<i>d</i>) is one if and only if d ∈ { − 1 , − 2 , − 3 , − 7 , − 11 , − 19 , − 43 , − 67 , − 163 } . {\displaystyle d\in \{\,-1,-2,-3,-7,-11,-19,-43,-67,-163\,\}.} <p>These are known as the <a href="/facts/Heegner_number/ExPvl9UM">Heegner numbers</a>. </p><p>By replacing d with the <a href="/facts/Discriminant_of_an_algebraic_number_field/mvv6MWKj">discriminant</a> D of Q(√<i>d</i>) this list is often written as: </p> D ∈ { − 3 , − 4 , − 7 , − 8 , − 11 , − 19 , − 43 , − 67 , − 163 } . {\displaystyle D\in \{-3,-4,-7,-8,-11,-19,-43,-67,-163\}.}