Primitive element (finite field)

<h2 id="properties">Properties</h2>
<h3>Number of primitive elements</h3>
<p>The number of primitive elements in a finite field GF(<i>q</i>) is <i>φ</i>(<i>q</i> − 1), where <i>φ</i> is <a href="/facts/Euler%27s_totient_function/mIQwXYzj">Euler's totient function</a>, which counts the number of elements less than or equal to <i>m</i> that are <a href="/facts/Coprime/bnCxCXxs">coprime</a> to <i>m</i>. This can be proved by using the theorem that the multiplicative group of a finite field GF(<i>q</i>) is <a href="/facts/Field_(mathematics)/xAjAS4ko">cyclic</a> of order <i>q</i> − 1, and the fact that a finite cyclic group of order <i>m</i> contains <i>φ</i>(<i>m</i>) generators.
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<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Simple_extension/xegn8Bpl">Simple extension</a></li>
<li><a href="/facts/Primitive_element_theorem/u5DvBoTl">Primitive element theorem</a></li>
<li><a href="/facts/Zech%27s_logarithm/AXPMg7WX">Zech's logarithm</a></li></ul>

<ul><li>Lidl, Rudolf; <a href="/facts/Harald_Niederreiter/MoayKG13">Harald Niederreiter</a> (1997). <a href="https://archive.org/details/finitefields0000lidl_a8r3"><i>Finite Fields</i></a> (2nd ed.). <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-39231-4.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/PrimitivePolynomial.html">"Primitive Polynomial"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>

Primitive element (finite field) open-in-new

Primitive element (finite field)