Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $q(n)$ denote the primitive $n$th roots of unity and let $K=\mathbb{Q}(q(n))$ be the associated cyclotomic field. Let $a$ denote the trace of $q(n)$ from $K$ to $\mathbb{Q}.$

How to

prove that $a=1$ if $n=1$, $a=0$ if $n$ is divisible by the square of a prime, and $a=(-1)^r$ if $n$ is the product of $r$ distinct primes?

share|improve this question
    
There is no reason to ever use all capital letters, even for a title. It is considered an extremely rude practice on the internet. Please don't do it again. –  Zev Chonoles Mar 25 '12 at 3:56
1  
Also, please explain what your thoughts are about your problem so far - are you stuck somewhere in particular? To do so is polite, and helps others better help you. –  Zev Chonoles Mar 25 '12 at 3:58
add comment

1 Answer 1

You are asked to compute the sum of all primitive $n$th roots of unity. Hints:

  1. What would you get if you computed the sum of all (not necessarily primitive) $n$th roots of unity?

  2. If an $n$th root of unity is not primitive, then it is a primitive $d$th root of unity for some divisor $d$ of $n$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.