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Let $K$ be a local field. Let $n$ be an arbitrary natural number and $e$ be any divisor of $n$.

Question Does there exist an extension $L/K$ with the following properties?

  1. $L/K$ is a cyclic extension, i.e. it is Galois with cyclic Galois group,
  2. $L/K$ has degree $n$ and
  3. $L/K$ has ramification index $e$.

My actual problem is it to find any cyclic extension $L/K$ with a fixed degree which is not totally ramified to construct so-called unramified characters of this fixed degree, but this question seems to be more interesting. I though about taking some cyclic extension $L'/K$ (if that's possible) and then take an intermediate subextension which must be cyclic due to the Fundamental Theorem of Galois Theory. But this approach does not resolve the problem with the degree resp. ramification indices.

Could you please help me with that problem? Thank you!

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This is not always possible. Let's stick to the totally ramified case $n=e$ to see this.
The theory of higher ramification groups poses some restrictions on the degree of totally ramified Galois extensions (cyclic or not). Let $q=p^n$ be the cardinality of the residue field $\kappa$ of $K$. Then for any Galois extension $L/K$ we have a filtration of $G=\mathrm{Gal}(L/K)$ given by the (higher) ramification groups $G \supset G_0 \supset G_1\supset \dots $. In the case that $L/K$ is totally ramified, we have $G=G_0$. One can show that $G_0/G_1$ injects into $\kappa^\times$, hence the order divides $q-1$ and for $i \geq 1$ $G_i/G_{i+1}$ injects into $\kappa$, hence the order is a power of $p$. Thus in this case the order of $\mathrm{Ga}(L/K)$ is of the form $u \cdot p^k$ where $u$ divides $q-1$. Thus it is not possible to have a totally ramified Galois extension of $K$ when the order has prime divisor that is not $p$ and doesn't divide $q-1$. For details regarding ramification groups, see these notes on algebraic number theory, especially lemma 9.1.5 (p. 193) and lemma 9.3.7 (p. 200).

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To avoid misunderstanding, let me recap what could be called your "hypothesis (H)". Let $K$ be a local field (you don't specify what this means, but for convenience, take $K$ to be an $l$-adic local field, i.e. a finite extension of $\mathbf Q_l$, so that the residue field will be a finite extension $\mathbf F_q$ of the prime field $\mathbf F_l$ ). For any integer $n=ef, e \neq 1$, does there exist a cyclic extension $L/K$ with degree $n$ and ramification index $e$ ? Stated as this, I'm afraid that hypothesis (H) does not hold true in general. To construct counter-examples, the main tool of course will be CFT (local here) which is supposed to describe completely the abelian extensions of $K$.

Counter-example. Fix a prime $p$ and consider hypothesis (H) for the abelian $p$-extensions of $K$. It will be convenient to introduce the maximal abelian pro-$p$-extension $\mathfrak K$ of $K$, with Galois group $\mathfrak G$. Recall that a $p$-extension $L/K$ is a finite Galois extension of degree equal to a power of $p$, and $\mathfrak K$ is the compositum of all such abelian $L$'s, so that $\mathfrak G$ is the projective limit of all such abelian $Gal(L/K)$'s. The structure of the $\mathbf Z_p$-module $\mathfrak G$ is classically known by CFT (see e.g. Cassels-Fröhlich, chap. XIV, thm. 1). We must distinguish two cases :

1) The tame case $p\neq l$: then $\mathfrak G \cong \mu(K) \times \mathbf Z_p $, where $\mu(K)$ is the group of $p$-primary roots of unity contained in $K$, and $\mathbf Z_p \cong $ the Galois group over $K$ of the maximal unramified pro-$p$-extension of $K$, topologically generated by the Frobenius automorphism at $p$ (beware of the mixture of multiplicative and additive notations). Note that $\mu(K)= 1$ iff $q \neq 1$ mod $p$. Hypothesis (H) for abelian $p$-extensions of $K$ in this case is obviously irrelevant. If $\mu(K)\neq 1$, the above decomposition of $\mathfrak G$ implies that the ramification index $e$ in hypothesis (H) is bounded by the maximal power of $p$ dividing $q-1$. This already shows that (H) cannot hold in general.

2) The wild case $p=l$ : We still have a decomposition $\mathfrak G \cong \mu(K) \times\mathbf Z_p^{1+N}$, where $N=[K:\mathbf Q_p]$, but the structural properties are naturally more complicated and are given by cup-products between generators (Demushkin's thm., see loc. cit.). One of the factors $\mathbf Z_p$ corresponds to the maximal pro-$p$-unramified extension of $K$, but the other factors can contribute to $e$. To go further, one needs to know more precisely what you are after ./.

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