# Find the sum of the 5th roots of unity?

Mar 7, 2018

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \text{the sum of the" \ \ 5^{"th"} \ \ "roots of unity is:} \setminus q \quad 0.$

#### Explanation:

$\text{By definition, the" \ \ 5^{"th"} \ \ "roots of unity are the solutions of the}$
$\text{equation:}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \quad {x}^{5} \setminus = \setminus 1. \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \left(I\right)$

$\text{And so they are also the solutions of the equation:}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \quad {x}^{5} - 1 \setminus = \setminus 0. \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \left(I I\right)$

$\text{However, the sum of the roots of the eqn. (II), as with any}$
$\text{monic polynomial, is the opposite of the coefficient of the}$
$\text{next-to-leading term. In eqn. (II), the next-to-leading term}$
$\text{is the" \ \ x^4 \ \ "term. Its coefficient is clearly" \ \ 0. \ \ "So its opposite is}$
$\text{clearly" \ \ 0. \ \ "And thus the sum of the solutions of (II) is} \setminus \setminus 0. \setminus \setminus$
$\text{Thus the sum of the solutions of (I) is" \ \ 0. \ \ "Thus:}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \text{the sum of the" \ \ 5^{"th"} \ \ "roots of unity is:} \setminus q \quad 0.$

$\text{Some Additional Remarks.}$

$\text{1) The above argument, thankfully trim, additionally shows:}$

$\setminus q \quad \setminus \quad \text{the sum of the" \ \ n^{"th"} \ \ "roots of unity is:" \qquad 0; \qquad "for} \setminus \setminus n \ge 2.$

$\setminus q \quad \setminus \quad \text{the sum of the" \ \ 1^{"st"} \ \ "roots of unity is:" \qquad \ 1; \qquad "trivially.}$

$\text{2) The product of the roots of any monic polynomial of}$
$\text{degree" \ \ n, "is" \ \ (-1)^n ( "constant term" ). \ \ "Applying this result to}$
$\text{the above polynomial, gives:}$

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

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