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Galois theory … of the Euler’s totient function

Let n>2 and let \omega \in \mathbb{C} be an n-th primitive root of unity. Prove that

    \[[\mathbb{Q}(\omega + \omega^{-1}) :\mathbb{Q}]=\frac{\varphi (n)}{2}\]

where \phi denotes the Euler’s totient function.

Solution

We have

    \[\phi(n)=[\mathbb{Q}(\omega):\mathbb{Q}]=[\mathbb{Q}(\omega):\mathbb{Q}(\omega +\omega^{-1})][\mathbb{Q}(\omega +\omega^{-1}):\mathbb{Q}]\]

where [\mathbb{Q}(\omega):\mathbb{Q}(\omega +\omega^{-1})]=2 since t^2-(\omega+\omega^{-1})t+1 is the minimal polynomial of \omega over \mathbb{Q}(\omega+\omega^{-1}).

The result follows.

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