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Galois extension
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a Galois extension is an <a href="/facts/Algebraic_extension/GAs76USY">algebraic</a> <a href="/facts/Field_extension/L7yIDtyK">field extension</a> <i>E</i>/<i>F</i> that is <a href="/facts/Normal_extension/HCIHmH4t">normal</a> and <a href="/facts/Separable_extension/9KBvXmz8">separable</a>; or equivalently, <i>E</i>/<i>F</i> is algebraic, and the <a href="/facts/Fixed_field/yEXwojME">field fixed</a> by the <a href="/facts/Automorphism_group/65vImKwe">automorphism group</a> Aut(<i>E</i>/<i>F</i>) is precisely the base <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> <i>F</i>. The significance of being a Galois extension is that the extension has a <a href="/facts/Galois_group/A0p35p5c">Galois group</a> and obeys the <a href="/facts/Fundamental_theorem_of_Galois_theory/srpbXtxW">fundamental theorem of Galois theory</a>. </p><p>A result of <a href="/facts/Emil_Artin/rI3vIFo2">Emil Artin</a> allows one to construct Galois extensions as follows: If <i>E</i> is a given field, and <i>G</i> is a finite group of automorphisms of <i>E</i> with fixed field <i>F</i>, then <i>E</i>/<i>F</i> is a Galois extension. </p><p>The property of an extension being Galois behaves well with respect to <a href="/facts/Composite_field_(mathematics)/jvIHa73A"> field composition and intersection</a>. </p>