Is it true that the splitting field for $x^n-1$ over $\mathbb{Q}$ is $\mathbb{Q}(\xi_n)$ where $\xi_n$ is a primitive n$^{th}$ root of unity, making it an extension of degree $\phi(n)$ (Euler phi function)? Every element of this extension should look like $\sum a_k\xi_n^k$?
2026-03-28 08:42:15.1774687335
splitting field of $x^n-1$ over $\mathbb{Q}$
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Yes, everything you said is true. I have only added that the $a_k$ should be elements in $\mathbb Q$. These fields are known as the Ciclotomic Fields.