grandes-ecoles 2019 Q7

grandes-ecoles · France · x-ens-maths1__mp Number Theory Algebraic Number Theory and Minimal Polynomials

Show that for all $n \geq 1$ we have $$X ^ { n } - 1 = \prod _ { d \mid n } \Phi _ { d }$$ the product being taken over the set of positive integers $d$ dividing $n$, where $\Phi_n = \prod_{z \in \mathbb{P}_n}(X - z)$ and $\mathbb{P}_n$ is the set of primitive $n$-th roots of unity.

Show that for all $n \geq 1$ we have
$$X ^ { n } - 1 = \prod _ { d \mid n } \Phi _ { d }$$
the product being taken over the set of positive integers $d$ dividing $n$, where $\Phi_n = \prod_{z \in \mathbb{P}_n}(X - z)$ and $\mathbb{P}_n$ is the set of primitive $n$-th roots of unity.

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