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You can view its source [e]" accesskey="e"><span>View source</span></a></li> <li id="ca-history" class="collapsible"><a href="/index.php?title=Cyclotomic_polynomials&action=history" title="Past revisions of this page [h]" accesskey="h"><span>History</span></a></li> </ul> </div>   <div id="p-cactions" class="springereomMenu emptyPortlet"> <h5><span>Actions</span><a href="#"></a></h5> <div class="menu"> <ul> </ul> </div> </div>  </div> </div> <div id="InnerContent"> <div id="mw-js-message" style="display:none;"></div>  <h1 id="firstHeading" class="firstHeading">Cyclotomic polynomials</h1>   <div id="bodyContent">  <div id="siteSub">From Encyclopedia of Mathematics</div>   <div id="contentSub"></div>   <div id="jump-to-nav"> Jump to: <a href="#mw-head">navigation</a>, <a href="#p-search">search</a> </div>   <div id="mw-content-text" lang="en" dir="ltr" class="mw-content-ltr"><div class="mw-parser-output"><p><br /> <i>circular polynomials</i> </p><p>The polynomials $ \Phi _ {1} , \Phi _ {2}, \dots $ that satisfy the relation </p><p>$$ x ^ {n} - 1 = \prod _ {d \mid n } \Phi _ {d} ( x), $$ </p><p>where the product is taken over all positive divisors $ d $ of the number $ n $, including $ n $ itself. Over the field of complex numbers one has </p><p>$$ \Phi _ {n} ( x) = \ \prod _ \zeta ( x - \zeta ), $$ </p><p>where $ \zeta $ ranges over the primitive $ n $-th roots of unity (cf. <a href="/wiki/Primitive_root" title="Primitive root">Primitive root</a>). The degree of $ \Phi _ {n} ( x) $ is the number of integers $ k $ among $ 1, \dots, n $ that are relatively prime with $ n $. The polynomials $ \Phi _ {n} ( x) $ can be computed recursively by dividing the polynomial $ x ^ {n} - 1 $ by the product of all $ \Phi _ {d} ( x) $, $ d < n $, $ d \mid n $. The coefficients lie in the prime field $ \mathbf Q $; in case of characteristic zero, they are integers. Thus, </p><p>$$ \Phi _ {1} ( x) = x - 1 ,\ \Phi _ {2} ( x) = x + 1 ,\ \ \Phi _ {3} ( x) = x ^ {2} + x + 1 . $$ </p><p>If, moreover, $ n= p $ is prime, then </p><p>$$ \Phi _ {p} ( x) = \frac{x ^ {p} - 1 }{x - 1 } = x ^ {p - 1 } + \dots + x + 1 . $$ </p><p>The polynomial $ \Phi _ {n} ( x) $ can be explicitly expressed using the <a href="/wiki/M%C3%B6bius_function" title="Möbius function">Möbius function</a> $ \mu $: </p><p>$$ \Phi _ {n} ( x) = \prod _ {d \mid n } ( x ^ {d} - 1 ) ^ {\mu ( n / d ) } . $$ </p><p>For example, </p><p>$$ \Phi _ {12} ( x) = $$ </p><p>$$ = \ ( x ^ {12} - 1) ( x ^ {6} - 1) ^ {-1} ( x ^ {4} - 1) ^ {-1} ( x ^ {3} - 1) ^ {0} ( x ^ {2} - 1) ( x - 1) ^ {0 } = $$ </p><p>$$ = \ \frac{x ^ {6} + 1 }{x ^ {2} + 1 } = x ^ {4} - x ^ {2} + 1 . $$ </p><p>All the polynomials $ \Phi _ {n} ( x) $ are irreducible over the field of rational numbers, but they may be reducible over finite prime fields. Thus, the relation </p><p>$$ \Phi _ {12} ( x) = x ^ {4} - x ^ {2} + 1 = ( x ^ {2} + 5x + 1 ) ( x ^ {2} - 5x + 1 ) $$ </p><p>is valid over the field of residues modulo 11. </p><p>The equation $ \Phi _ {n} ( x) = 0 $, which gives all primitive $ n $-th roots of unity, is known as the equation of division of the circle. The solution of this equation in trigonometric form is </p><p>$$ r _ {k} = \cos \frac{2 k \pi }{n} + i \sin \frac{2 k \pi }{n} = \ e ^ {2 k \pi i /n } , $$ </p><p>where the fraction $ k / n $ is irreducible, i.e. $ k $ and $ n $ are relatively prime. The solution of the equation of division of the circle by radicals is closely connected with the problem of constructing a regular $ n $-gon, or with the equivalent problem of subdividing the circle into $ n $ equal parts; the latter problem can be solved using a straightedge and a pair of compasses if and only if the equation $ \Phi _ {n} ( x) = 0 $ is solvable in quadratic radicals. It was shown by C.F. Gauss in 1801 that this condition is satisfied if and only if </p><p>$$ n = 2 ^ {m} p _ {1} \cdots p _ {s} , $$ </p><p>where $ m $ is a non-negative integer and $ p _ {1}, \dots, p _ {s} $ are pairwise different prime numbers of the form $ 2 ^ {2 ^ {k} } + 1 $, where $ k $ is a non-negative integer. </p> <h4><span class="mw-headline" id="References">References</span></h4> <table><tbody><tr><td valign="top">[1]</td> <td valign="top"> B.L. van der Waerden, "Algebra" , <b>1–2</b> , Springer (1967–1971) (Translated from German)</td></tr><tr><td valign="top">[2]</td> <td valign="top"> A.K. Sushkevich, "Foundations of higher algebra" , Moscow-Leningrad (1941) (In Russian)</td></tr></tbody></table> <h4><span class="mw-headline" id="Comments">Comments</span></h4> <p>The degree of $ \Phi _ {n} ( x) $ is equal to $ \phi ( n) $, the value (at $ n $) of the Euler $ \phi $-function (cf. <a href="/wiki/Euler_function" title="Euler function">Euler function</a>). </p><p>Prime numbers of the form $ 2 ^ {2 ^ {k} } + 1 $ with $ k $ a non-negative integer are called Fermat primes, these numbers are related to a problem of Fermat: When is the number $ F _ {k} = 2 ^ {2 ^ {k} } + 1 $, with $ k $ as before, prime? The only small Fermat primes are $ F _ {0} $, $ F _ {1} $, $ F _ {2} $, $ F _ {3} $, $ F _ {4} $ (cf. <a href="#References">[a1]</a>, pp. 183-185). </p> <h4><span class="mw-headline" id="References_2">References</span></h4> <table><tbody><tr><td valign="top">[a1]</td> <td valign="top"> I. Stewart, "Galois theory" , Chapman & Hall (1973)</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> H. Davenport, "Multiplicative number theory" , Springer (1980)</td></tr></tbody></table>    </div><div id="citeRevision" class="springerCitation"><strong>How to Cite This Entry:</strong><br /> Cyclotomic polynomials. <i>Encyclopedia of Mathematics.</i> URL: http://encyclopediaofmath.org/index.php?title=Cyclotomic_polynomials&oldid=52204</div><div id="originalIsbn" class="springerCitation">This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. <a href="http://encyclopediaofmath.org/index.php?title=Cyclotomic_polynomials&oldid=16160">See original article</a></div></div>   <div class="printfooter"> Retrieved from "<a dir="ltr" href="https://encyclopediaofmath.org/index.php?title=Cyclotomic_polynomials&oldid=52204">https://encyclopediaofmath.org/index.php?title=Cyclotomic_polynomials&oldid=52204</a>" </div>   <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:Categories" title="Special:Categories">Categories</a>: <ul><li><a href="/wiki/Category:TeX_auto" title="Category:TeX auto">TeX auto</a></li><li><a href="/wiki/Category:TeX_done" title="Category:TeX done">TeX done</a></li></ul></div></div>  <div class="visualClear"></div>   </div>  </div> </div> </div> <div id="footer"> <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 7 March 2022, at 05:18.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="/wiki/Encyclopedia_of_Mathematics:Privacy_policy" title="Encyclopedia of Mathematics:Privacy policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Encyclopedia_of_Mathematics:About" class="mw-redirect" title="Encyclopedia of Mathematics:About">About Encyclopedia of Mathematics</a></li> <li id="footer-places-disclaimer"><a href="/wiki/Encyclopedia_of_Mathematics:General_disclaimer" title="Encyclopedia of Mathematics:General disclaimer">Disclaimers</a></li> </ul> <ul id="footer-icons"> <li><a href="/index.php/Encyclopedia_of_Mathematics:Copyrights">Copyrights</a> </li> <li> <a href="/index.php/Encyclopedia_of_Mathematics:Legal">Impressum-Legal</a></li> </ul> <div style="clear:both"></div> <span class="manage-cookies"><a class="optanon-show-settings">Manage Cookies</a></span> </div>  <script type="<br /> <b>Warning</b>: Undefined array key "jsmimetype" in <b>/var/www/html/includes/skins/QuickTemplate.php</b> on line <b>99</b><br /> "> if ( window.isMSIE55 ) fixalpha(); </script>   <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgPageParseReport":{"limitreport":{"cputime":"0.008","walltime":"0.010","ppvisitednodes":{"value":53,"limit":1000000},"postexpandincludesize":{"value":126,"limit":2097152},"templateargumentsize":{"value":8,"limit":2097152},"expansiondepth":{"value":5,"limit":40},"expensivefunctioncount":{"value":0,"limit":100},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":0,"limit":5000000},"timingprofile":["100.00% 1.543 1 -total"," 95.33% 1.471 2 Template:TEX"]},"cachereport":{"timestamp":"20241203220517","ttl":86400,"transientcontent":false}}});mw.config.set({"wgBackendResponseTime":66});});</script></div> </div> </body> </html>