Question

Roots of unity for root n form a group with multiplication proof

Answer 1

Answer:

Yes

Explanation:

Identity: #1# is the identity.

Inverse: If #a# is an #n#th root of unity, then so is #1/a#, since:

#(1/a)^n = 1/(a^n) = 1/1 = 1#

Closure under product: If #a# is an #m#th root of unity and #b# an #n#th root of unity, then #ab# is an #mn#th root of unity:

#(ab)^(mn) = (a^m)^n(b^n)^m = 1^n*1^m = 1*1 = 1#

Associativity: Inherited from the complex numbers:

#a(bc) = (ab)c " "# for any #a, b, c#

#color(white)()#

Footnote

The elements of this group are all the numbers of the form:

#cos theta + i sin theta#

where #theta# is a rational multiple of #pi#.