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Let $\zeta$ be the $p^l$-th root of unity in the complex plane, where $p$ is a prime number.

Suppose $M$ be a finitely generated $\mathbb{Q}(\zeta)$-module.

Is it true that $\dim_\mathbb{Q} M\geq \frac{1}{p^l} \dim_{\mathbb{Q}(\zeta)} M$?

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up vote 1 down vote accepted

The field extension is of degree $[\mathbb Q(\zeta):\mathbb Q]=\phi(p^l)=p^{l-1}(p-1)$, therefore we obtain $$ \dim_{\mathbb Q(\zeta)} M = p^{l-1}(p-1) \dim_{\mathbb Q} M $$ for the vector space dimensions.

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