The cyclotomic extension $K({\mu }_n)$ as a splitting field of $t^n-1$

Let $K$ be a field and $n\ge 1$. A cyclotomic extension of $K$ of order $n$ is a splitting field $E$ of $t^n-1$ over $K$ (Polynomials that split and splitting fields of a polynomial or a family of polynomials); one exists by Every nonzero polynomial over a field has a splitting field. It is written

$K({\mu }_n):=E\mathrm{.}$

The notation is accurate. The roots of $t^n-1$ in $E$ are exactly the elements of ${\mu }_n(E)$ (The group ${\mu }_n(K)$ of $n$-th roots of unity in a field, and primitive $n$-th roots of unity), and a splitting field is generated over $K$ by the roots, so

$E=K({\mu }_n(E))$

in the sense of Finitely generated field extensions $F(a_1,\dots ,a_r)$: $E$ is the smallest subfield of itself containing $K$ and the $n$-th roots of unity it holds.

When the characteristic does not divide $n$ the extension has a single generator: by $t^n-1$ is separable over $K$ exactly when the characteristic does not divide $n$, and then a splitting field carries $n$ distinct $n$-th roots of unity the group ${\mu }_n(E)$ is then cyclic of order $n$ and

$K({\mu }_n)=K(\zeta )$

for any primitive $n$-th root of unity $\zeta \in E$. Without that hypothesis the notation still names a splitting field, but ${\mu }_n(E)$ can be much smaller than $n$ and no primitive $n$-th root of unity need exist (In characteristic $p$ the only $p^k$-th root of unity is $1$, and $t^{p^k}-1=(t-1{)}^{p^k}$).

• Which splitting field. Any two splitting fields of $t^n-1$ over $K$ are $K$-isomorphic (Any two splitting fields of a polynomial are isomorphic over the base field), and every statement made below about $K({\mu }_n)$ is invariant under a $K$-isomorphism, so the definite article is harmless; where a fixed ambient field matters, as in the compositum and intersection results, the statement says so and works inside one chosen extension of $K$.