# "By definition, the" \ \ 5^{"th"} \ \ "roots of unity are the solutions of the" #

# "equation:" #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad x^5 \ = \ 1. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ (I) #

# "And so they are also the solutions of the equation:" #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad x^5 - 1 \ = \ 0. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (II) #

# "However, the sum of the roots of the eqn. (II), as with any" #

# "monic polynomial, is the opposite of the coefficient of the" #

#"next-to-leading term. In eqn. (II), the next-to-leading term" #

# "is the" \ \ x^4 \ \ "term. Its coefficient is clearly" \ \ 0. \ \ "So its opposite is" #

# "clearly" \ \ 0. \ \ "And thus the sum of the solutions of (II) is" \ \ 0. \ \ #

# "Thus the sum of the solutions of (I) is" \ \ 0. \ \ "Thus:" #

# \qquad \qquad \qquad \qquad "the sum of the" \ \ 5^{"th"} \ \ "roots of unity is:" \qquad 0. #

# "Some Additional Remarks." #

# "1) The above argument, thankfully trim, additionally shows:" #

# \qquad \quad "the sum of the" \ \ n^{"th"} \ \ "roots of unity is:" \qquad 0; \qquad "for" \ \ n >= 2. #

# \qquad \quad "the sum of the" \ \ 1^{"st"} \ \ "roots of unity is:" \qquad \ 1; \qquad "trivially." #

# "2) The product of the roots of any monic polynomial of" #

# "degree" \ \ n, "is" \ \ (-1)^n ( "constant term" ). \ \ "Applying this result to" #

# "the above polynomial, gives:" #

# "the product of the" \ \ n^{"th"} \ \ "roots of unity is:" \qquad -1; \qquad "for" \ \ n \quad "odd"; #

# "the product of the" \ \ n^{"th"} \ \ "roots of unity is:" \qquad +1; \qquad "for" \ \ n \quad "even". #