# Globalizing local field extensions with controlled ramification

Let $K_1/k_1, \ldots, K_r/k_r$ be "separable cyclic" extensions of degree $n$ where each $k_i$ is a local field of characteristic 0 (archimedean or not). By separable cyclic of degree $n$, I mean the localization of some global cyclic extension of degree $n$, so a given $K_i/k_i$ may be split or not. (E.g., if $n=2$, either $K_i/k_i$ is a quadratic field extension or $K_i = k_i \oplus k_i$.)

Does there exist an extension of number fields $E/F$ globalizing all $K_i/k_i$ and unramified everywhere else? That is, is there a global $E/F$ with a set $S = \{v_1, \ldots v_r \}$ of places of $k$ such that $E_{v_i} = K_i$, $F_{v_i} = k_i$ and $E_v/F_v$ is unramified for $v \not \in S$.

Results are known about globalizing local field extensions when one fixed a global field $F$, e.g. Grunwald-Wang, but I don't want to do that. Indeed, a positive answer is not always possible if one, say, requires $F=\mathbb Q$, even if each $F_{v_i}$ specified is a distinct local completion of $\mathbb Q$. (For instance, take $n=2$, specify that $E$ should be $\mathbb C$ at infinity, ramified at 3 and unramified everywhere else---there is a unique choice globally $E=\mathbb Q(\sqrt{-3})$, but locally there are 2 ramified extensions at 3 one can choose.)

Rather I want to prescribe stronger local conditions than what one gets from Grunwald-Wang. A positive answer (even with some conditions) should allow one to use global methods to prove results about local representation theory via trace formulas.

Note this is related to finding number fields $F$ such that a ray class field has prescribed local behaviour.

I imagine such questions are hard, so partial results are welcome. (I'm not even sure whether the answer is yes when $n=2$ and $r=1$, though I think it is doable.)

Edit: Based on the answer and comments, I want to clarify that when $r=1$, the question is asking for a global extension $E/F$ which is unramified at all but 1 place, where it is $K_1/k_1$. The current comments and answer seem to give an extension $E/F$ that at all ramified places $v \in R$ looks like $K_1/k_1$, but at least a priori we do not have $|R| = 1$.

• If you fix and $F$, then it is very easy, using class field theory, to construct and $E/F$ with all the required properties for primes in $S$ but possibly with additional ramification at some auxilary set $T$. But then one simply constructs an extension $F'/F$ which is sufficiently ramified at $T$, totally split at $S$, and totally disjoint from $E/F$ (this is also easy by class field theory). Then take the extension $E'/F'$, where $E' = E.F'$. May 21, 2016 at 15:23
• What do you mean by " the answer to this question is clearly false"? That your question has a negative answer? Then what exactly do you want to know? Jun 18, 2016 at 9:09
• @FranzLemmermeyer Sorry, I meant to say this is not always possible if one fixed a base global field. I revised that paragraphs. Does it make sense now? Jun 18, 2016 at 9:22
• Does this mean that Gypsum's comment completely answers your question? Jun 18, 2016 at 10:11
• @FranzLemmermeyer I think when Gypsum passes from E/F to E'/F' (say degree n) one replaces S with some S' (which is basically n copies of S), correct? For instance if I just wanted $E/F$ to be ramified at a single place (with specified local completions $E_{w_0}/F_{v_0}$), then Gypsum's construction doesn't necessarily do this---it only gives an extension which looks like $E_{w_0}/F_{v_0}$ at several places and is unramified everywhere else, no? Jun 18, 2016 at 11:36

• Thanks for the answer. However, and I'm sorry if it wasn't clear, but I want to do things like given some specific ramified local extension $E/F$ get a global field $K/k$ which is only ramified at a single place $v_1$, where it is isomorphic to $E/F$. It seems to me your note can give me a global extension $K/k$ which is isomorphic to $E/F$ at $r$ distinct places and unramified elsewhere, but without control on $r$. Is this correct? May 1, 2018 at 15:15