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Local class field theory
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, local class field theory, introduced by <a href="/facts/Helmut_Hasse/WFtCX2jI">Helmut Hasse</a>, is the study of <a href="/facts/Abelian_extension/vIdW4Hnv">abelian extensions</a> of <a href="/facts/Local_field/vqIgPIX3">local fields</a>; here, "local field" means a field which is complete with respect to an <a href="/facts/Absolute_value_(algebra)/UUncm7y0">absolute value</a> or a <a href="/facts/Discrete_valuation/Q54hIUDJ">discrete valuation</a> with a finite <a href="/facts/Residue_field/e6nflEmB">residue field</a>: hence every local field is <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> (as a <a href="/facts/Topological_field/qxQqzhCy">topological field</a>) to the <a href="/facts/Real_numbers/R02gw5Pb">real numbers</a> R, the <a href="/facts/Complex_numbers/2w7mqI7e">complex numbers</a> C, a <a href="/facts/Finite_extension/hY5LgHYk">finite extension</a> of the <a href="/facts/P-adic_number/SDgDbkLF"><i>p</i>-adic numbers</a> Q<i>p</i> (where <i>p</i> is any <a href="/facts/Prime_number/L20FLkgn">prime number</a>), or the field of <a href="/facts/Formal_Laurent_series/CNmaG0Vk">formal Laurent series</a> F<i>q</i>((<i>T</i>)) over a <a href="/facts/Finite_field/UkMGJo3m">finite field</a> F<i>q</i>. </p>