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Golden field
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<p>In mathematics, Q ( 5 ) {\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} , sometimes called the golden field, is the <a href="/facts/Real_quadratic_field/dUnSDN4e">real quadratic field</a> obtained by <a href="/facts/Field_extension/L7yIDtyK">extending</a> the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> with the <a href="/facts/Square_root_of_5/H3mUJD8Q">square root of 5</a>. The elements of this field are all of the numbers a + b 5 {\displaystyle a+b{\sqrt {5}}} , where a {\displaystyle a} and b {\displaystyle b} are both rational numbers. As a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>, Q ( 5 ) {\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} supports the same basic <a href="/facts/Arithmetic/hvHwkbMV">arithmetical operations</a> as the rational numbers. The name comes from the <a href="/facts/Golden_ratio/anloSRmD">golden ratio</a> φ = 1 2 ( 1 + 5 ) {\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}} , which is the <a href="/facts/Fundamental_unit_(number_theory)/T3dJERRQ">fundamental unit</a> of Q ( 5 ) {\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}} , and which satisfies the equation φ 2 = φ + 1 {\displaystyle \textstyle \varphi ^{2}=\varphi +1} . </p><p>Calculations in the golden field can be used to study the <a href="/facts/Fibonacci_sequence/mqxpAUQD">Fibonacci sequence</a> and other topics related to the golden ratio, notably the <a href="/facts/Geometry/wTQf5ffQ">geometry</a> of the <a href="/facts/Regular_pentagon/xoq5oJeQ">regular pentagon</a> and higher-dimensional shapes with fivefold <a href="/facts/Dihedral_symmetry/1PUlRh4M">symmetry</a>. </p>