Editing Angle trisection (section)

== Angles which can be trisected ==
However, some angles can be trisected. For example, for any [[constructible number|constructible]] angle {{math|''θ''}}, an angle of measure {{math|3''θ''}} can be trivially trisected by ignoring the given angle and directly constructing an angle of measure {{math|''θ''}}.  There are angles that are not constructible but are trisectible (despite the one-third angle itself being non-constructible).  For example, {{math|{{sfrac|3{{pi}}|7}}}} is such an angle: five angles of measure {{math|{{sfrac|3{{pi}}|7}}}} combine to make an angle of measure {{math|{{sfrac|15{{pi}}|7}}}}, which is a full circle plus the desired {{math|{{sfrac|{{pi}}|7}}}}.

For a [[positive integer]] {{mvar|N}}, an angle of measure {{math|{{sfrac|2{{pi}}|''N''}}}} is ''trisectible'' if and only if {{math|3}} does not divide {{mvar|N}}.<ref>MacHale, Desmond. "Constructing integer angles", ''Mathematical Gazette'' 66, June 1982, 144–145.</ref><ref name=McLean>{{cite journal |author=McLean, K. Robin |title=Trisecting angles with ruler and compasses |journal=Mathematical Gazette |volume=92 |date=July 2008 |pages=320–323 |doi=10.1017/S0025557200183317 |s2cid=126351853 |url=https://www.cambridge.org/core/journals/mathematical-gazette/article/9252-trisecting-angles-with-ruler-and-compasses/FD6933F81AC55AF2225AF75568E2103E |quote=See also Feedback on this article in vol. 93, March 2009, p. 156.}}</ref>  In contrast, {{math|{{sfrac|2{{pi}}|''N''}}}} is ''constructible'' if and only if {{mvar|N}} is a power of {{math|2}} or the product of a power of {{math|2}} with the product of one or more distinct [[Fermat prime]]s.

===Algebraic characterization===
Again, denote the set of [[rational numbers]] by {{math|'''Q'''}}.

[[Theorem]]: An angle of measure {{math|''θ''}} may be trisected [[if and only if]] {{math|''q''(''t'') {{=}} 4''t''<sup>3</sup> − 3''t'' − cos(''θ'')}} is reducible over the [[field extension]]  {{math|'''Q'''(cos(''θ''))}}.

The [[Mathematical proof|proof]] is a relatively straightforward generalization of the proof given above that a {{math|60°}} angle is not trisectible.<ref name=Stewart>{{cite book | last = Stewart | first = Ian  | author-link = Ian Stewart (mathematician) | title = ''Galois Theory'' | publisher = Chapman and Hall Mathematics | year = 1989 | pages = g. 58 | isbn = 978-0-412-34550-0 | title-link = Galois Theory  }}</ref>

===Other numbers of parts===
For any nonzero integer {{mvar|N}}, an angle of measure {{math|{{frac|2{{pi}}|''N''}}}} radians can be divided into {{mvar|n}} equal parts with straightedge and compass if and only if {{mvar|n}} is either a power of {{math|2}} or is a power of {{math|2}} multiplied by the product of one or more distinct Fermat primes, none of which divides {{mvar|N}}. In the case of trisection ({{math|''n'' {{=}} 3}}, which is a Fermat prime), this condition becomes the above-mentioned requirement that {{mvar|N}} not be divisible by {{math|3}}.<ref name=McLean/>