Examples of cyclotomic extensions with Galois group not isomorphic to $U(\mathbb{Z}_n)$

Question

I'm working under the hypothesis that all the roots of polynomial $X^n - 1$ are different (so $Car(K) = 0 \lor p = Car(K) \nmid n$).

I was a bit surprised when I proved that $Gal(F/K)$ where $F/K$ is a cyclotomic extension has to be isomorphic to a subgroup of $U(\mathbb{Z}_n)$ while there are exactly $\phi(n)$ primitive roots of unity.

Therefore, my intuition tells me that some of the Galois automorphism are repeated or they are not possible by some reason (based on the characteristic).

Can anybody provide an explicit example of this situation?

Edit:

I would add to existing answers that it depends on the irreducibility of the polynomial since the orbits are the different irreducible factors.

Answer 1

No Galois automorphism is repeated. The problem is that the $n$-th cyclotomic polynomial, which is irreducible over $\mathbf Q$, might be reducible over a larger base field $K$, so some of the Galois automorphisms over $\mathbf Q$ might not make sense over a bigger field.

A simple example is base field $\mathbf R$, where irreducibles have degree at most $2$. Try $\mathbf R(\zeta_n)/\mathbf R$ for $n > 6$. The Galois group is the identity and complex conjugation (corresponding to $\pm 1\bmod n$ in the units mod $n$), which has size $2$, while $\varphi(n) > 2$ for $n > 6$.

Answer 2

Let $K=\mathbb{Q}(i)$, and let $F$ be the splitting field (over $K$) of $X^8-1$. Then $F=K(\sqrt{2})$, hence $F$ has degree $2$ over $K$, even though $\phi(8)=4$.