What is the value of $ \ n\ $ such that $\ \xi_{n}= e^{\frac{2 \pi i}{n}} \ $ has degree at most $ \ 3 \ $ i.e, $ [ \mathbb{Q}(\xi) : \mathbb{Q}]=3 $ . $$ $$ I have thought that - since $ [\mathbb{Q}(\xi_{n}) : \mathbb{Q}]=\phi(n) $ , we have $ \ \phi(n)=3 $ . Or , n= ? . please help me .
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Last time I checked $\Bbb{Q}$ had infinitely many elements, so I removed the tag [tag:finite-fields]. – Jyrki Lahtonen May 15 '17 at 04:48
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why to make negative grade – MAS May 15 '17 at 06:36
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Note that the question asks for degree at most 3. So possible answers include $n=2,3$. Also, notice that $\phi(n)$ is even for $n\ge 3$ (see here Why is Euler's Totient function always even?) so you cannot get exact equality.
helloworld112358
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14,6 are fine as well. It was not meant to be an exhaustive list – helloworld112358 May 15 '17 at 00:41