On the Stickelberger ideal and the relative class number
On the Stickelberger ideal and the relative class number
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"><mml:semantics><mml:mi>k</mml:mi><mml:annotation encoding="application/x-tex">k</mml:annotation></mml:semantics></mml:math></inline-formula>be any imaginary abelian field,<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"><mml:semantics><mml:mi>R</mml:mi><mml:annotation encoding="application/x-tex">R</mml:annotation></mml:semantics></mml:math></inline-formula>the integral group ring of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals Gal left-parenthesis k slash double-struck upper Q right-parenthesis"><mml:semantics><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext>Gal</mml:mtext></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Q</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">G = {\text {Gal}}(k/\mathbb {Q})</mml:annotation></mml:semantics></mml:math></inline-formula>, and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"><mml:semantics><mml:mi>S</mml:mi><mml:annotation encoding="application/x-tex">S</mml:annotation></mml:semantics></mml:math></inline-formula>the Stickelberger ideal …