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Galois representation
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a Galois module is a <a href="/facts/G-module/hglV6Hfv"><i>G</i>-module</a>, with <i>G</i> being the <a href="/facts/Galois_group/A0p35p5c">Galois group</a> of some <a href="/facts/Field_extension/L7yIDtyK">extension</a> of <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a>. The term Galois representation is frequently used when the <i>G</i>-module is a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over a field or a <a href="/facts/Free_module/o6rC6yoJ">free module</a> over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> in <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a>, but can also be used as a synonym for <i>G</i>-module. The study of Galois modules for extensions of <a href="/facts/Local_field/vqIgPIX3">local</a> or <a href="/facts/Global_field/na458xWs">global fields</a> and their <a href="/facts/Galois_cohomology/nzC1EVYT">group cohomology</a> is an important tool in <a href="/facts/Number_theory/zhoa7zrc">number theory</a>. </p>