Cyclotomic conductor of a full cyclotomic field

Let $n\ge 1$ and let $K=\mathbb{Q}({\mu }_n)$ be a cyclotomic extension of $\mathbb{Q}$ of order $n$ (The cyclotomic extension $K({\mu }_n)$ as a splitting field of $t^n-1$): a splitting field of $t^n-1$ over $\mathbb{Q}$. When a primitive $n$-th root of unity ${\zeta }_n$ is fixed, $K=\mathbb{Q}({\zeta }_n)$.

A positive integer $f$ is admissible for $K$ when there is a $\mathbb{Q}$-algebra embedding $K\hookrightarrow F$ into a splitting field $F$ of $t^f-1$ over $\mathbb{Q}$. The cyclotomic conductor of $K$ is the least admissible positive integer,

$\mathrm{cond}(K):=\min \{f\ge 1:K\hookrightarrow \mathbb{Q}({\zeta }_f)\}\mathrm{.}$

Well-definedness. The set is nonempty, since the identity of $K$ exhibits $K\hookrightarrow K$ and $K=\mathbb{Q}({\mu }_n)$ is a splitting field of $t^n-1$; by the well-ordering of the positive integers it therefore has a least element (The well-ordering principle). The value depends only on the $\mathbb{Q}$-isomorphism class of $K$: if $\phi :K\to K^{\mathrm{\prime }}$ is an isomorphism of splitting fields of $t^n-1$ and $K\hookrightarrow \mathbb{Q}({\zeta }_f)$ is an embedding, then composing with ${\phi }^{-1}$ exhibits $K^{\mathrm{\prime }}\hookrightarrow \mathbb{Q}({\zeta }_f)$, so $K$ and $K^{\mathrm{\prime }}$ have the same admissible integers. In particular the conductor is unchanged by the choice of splitting field (Any two splitting fields of a polynomial are isomorphic over the base field).

Conductors are compared inside a common field. Every assertion below about an inclusion $K\subseteq \mathbb{Q}({\zeta }_f)$ is read inside one fixed algebraic closure of $\mathbb{Q}$, in which one copy of each cyclotomic field has been chosen; by the previous paragraph this loses no information, because conductor statements are invariant under the $\mathbb{Q}$-isomorphisms relating the choices.

Scope. This is a conductor of a full cyclotomic field only. It is not the Artin conductor of a Dirichlet character, not a conductor assigned to an arbitrary abelian number field, and no Kronecker-Weber premise is used or implied: the definition does not assert that an arbitrary abelian field lies in some $\mathbb{Q}({\zeta }_f)$. Admissibility of $f=n$ does not make $n$ the least admissible integer; identifying the least one is the content of Conductor of a full cyclotomic field.