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Check when a positive square Root $\sqrt{d}$ is contained in Cyclotomic Field

The Galois group of $\mathbb Q(\zeta_n)/\mathbb Q$ is $G_n=(\mathbb Z/n\mathbb Z)^\times$. If $n=\prod_{i=1}^k p_i^{m_i}$ is the factorization of $n$, this gives us $G_n\cong \prod_{i=1}^k G_{p_i^{m_i}...
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Why should a splitting field of a polynomial over $\mathbb{F}_p[X]$ be a cyclotomic extension?

Since an extension of $\Bbb F_p$ of degree $n$ has order $p^n$ (it's a vector space of dimension $n$ over $\Bbb F_p$), we have that the multiplicative group has order $p^n-1.$ By Lagrange's theorem ...
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