Math Is Fun Forum

laipou

Member

Registered: 2009-09-19

Posts: 24

square free integer and root of unity

Which roots of unity are contained in

,where  is a square free integer?
Thanks for any help.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: square free integer and root of unity

d > 0 is easy.

For d < 0, we have that d^2 is in Q.  Thus, all polynomials can be represented by a + bd, where a and b are rationals.  Now you want to find all elements of this form z where z^n = 1.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

laipou

Member

Registered: 2009-09-19

Posts: 24

Re: square free integer and root of unity

Thank you for you suggestion,Ricky.
Here is the answer I think.
Let

be a primitive nth root of unity of

.
Then we know

(Euler function).
Since

,if

,then we must have

or

.
Finaly,we get

.
                   

.
                   

o,w.