a. Prove that if the regular mn-gon is constructible, then theregular m- and n-g

Question

a. Prove that if the regular mn-gon is constructible, then theregular m- and n-gons are constuctible as well. b. Prove that if gcd(m,n) = 1 and the regular m- and n-gonsare both contructible, then the regular mn-gon isconstructible. **PLEASE SHOW ALL WORK~~ a. Prove that if the regular mn-gon is constructible, then theregular m- and n-gons are constuctible as well. b. Prove that if gcd(m,n) = 1 and the regular m- and n-gonsare both contructible, then the regular mn-gon isconstructible. **PLEASE SHOW ALL WORK~~

Explanation / Answer

a. regular mn-gon is constructible means that the vertices areconstructible numbers. But the vertices (with canonicalchoice of center as the origin of the regular polygons) havecoordinates e^(2pi / mn) = cos (2pi / mn) + i sin (2pi / mn), andso this must be in a field L over Q of some 2^k degree. Clearly, then, e^(2pi/m) and e^(2pi/n) both exist in this L, sincethey are powers of e^(2pi / mn). So m- and n- gons are alsoconstructible. b. This amounts to showing that if 1 and2 are primitive m-th and n-th roots of unity,respectively, then 12 is a primitive mn-th root ofunity, if gcd(m,n) = 1. But this is just an easy fact fromgroup theory.

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