primitive roots, field dimension

Question

Let $\zeta$ be a primitive $m$-th root of $1$.

Determine the values of $m$ such that: $[\mathbb Q$($\zeta$):$\mathbb Q$]$=2$.

The only thing I have in mind is that $[\mathbb Q $($\zeta$):$\mathbb Q$]=$\varphi (n)$= Euler $\varphi$ fuction. But I am not sure if it has something to do with this question.

score 6 · Accepted Answer · answered Dec 30 '13 at 16:18

6

It's the way to do it. Now find all $n$ such that $\varphi(n) = 2$.

Hint 1.

If $p$ is a prime, and $p^2 \mid n$, then $p \mid \varphi(n)$.

Hint 2.

If $p$ is a prime, and $p \mid n$, then $p-1 \mid \varphi(n)$.

answered Dec 30 '13 at 16:18

Andreas Caranti

68,873

2

This is your $666^\text{th}$ answer and your hat fits you nicely. – Git Gud Dec 30 '13 at 16:27
@GitGud, you mean the number of the... :-P – Andreas Caranti Dec 30 '13 at 16:29
1

Ah, diabolic....+1 – DonAntonio Dec 30 '13 at 16:40

primitive roots, field dimension

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