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Reference.org
Algebraic function field
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an algebraic function field (often abbreviated as function field) of <i>n</i> variables over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> <i>k</i> is a finitely generated <a href="/facts/Field_extension/L7yIDtyK">field extension</a> <i>K</i>/<i>k</i> which has <a href="/facts/Transcendence_degree/ubxWrbjJ">transcendence degree</a> <i>n</i> over <i>k</i>. Equivalently, an algebraic function field of <i>n</i> variables over <i>k</i> may be defined as a <a href="/facts/Finite_field_extension/L7yIDtyK">finite field extension</a> of the field <i>K</i> = <i>k</i>(<i>x</i>1,...,<i>x</i><i>n</i>) of <a href="/facts/Rational_functions/2nJGu0Ko">rational functions</a> in <i>n</i> variables over <i>k</i>. </p>