I am asking [my question](https://math.stackexchange.com/questions/8332/roots-of-unity-and-field-extensions) here, since it's been voted up a fair bit on math.SE, but without answers, so it may be harder than I assumed it was.


Can we always break an arbitrary field extension $L/K$ into an extension $F/K$ in which the only roots of unity of $F$ are those in $K$, i.e. $\mu\_F=\mu\_K$, followed by an extension $L/F$ which is of the form $L=F(\{\omega\_i\})$, where the $\omega\_i$ are roots of unity? If not, are there reasonable hypotheses (e.g. separable, finite) on $L/K$ that would make this true?


My motivation was simply that the other order, i.e. breaking an arbitrary extension $L/K$ into one where $F=K(\{\omega\_i\})$ for some roots of unity $\omega\_i$, followed by $L/F$ where $\mu\_L=\mu\_K$, is obvious - specifically, set $F=K(\mu\_L)$.


Now, my first (naive) approach was to try to construct the "maximum" intermediate field which does not add roots of unity by taking the compositum of all such intermediate extensions. However, this doesn't exist even for number fields, e.g. setting $K=\mathbb{Q}$, $L=\mathbb{Q}(\zeta\_3,\sqrt[3]{2})$, $E\_1=\mathbb{Q}(\sqrt[3]{2})$, and $E\_2=\mathbb{Q}(\zeta\_3\sqrt[3]{2})$, we have $\mu\_{E\_1}=\mu\_{E\_2}=\mu\_K$, but $\mu\_{E\_1E\_2}=\mu\_{L}\supsetneq\mu\_K$.


Note that $K=\mathbb{Q}$ and $L=\mathbb{Q}(\zeta\_3,\sqrt[3]{2})$ isn't a counterexample to the actual problem - for example, $F=E\_1$ works, because $\mu\_{E\_1}=\mu\_{\mathbb{Q}}$, and $L=E\_1(\zeta\_3)$.


So, to prove the claim / construct a counterexample, it seems to me that we want to look at intermediate fields $E$ which are maximal among those such that $\mu\_E=\mu\_K$, and determine whether or not there always exists at least one such $E$ such that $L=E(\text{some roots of unity})$.




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Here is Arturo Magidin's comment on the original question:



> 
> not a proof/counterexample, but an observation: suppose $L$ is Galois over $K$; we can let $M$ be the extension of $K$ obtained by adding all roots of unity in $L$; this is Galois over $K$, so corresponds to a normal subgroup $H$ of $G=\text{Gal}(L/K)$. If we can break up the extension as you mention, then $L$ is Galois over $F$, and $\text{Gal}(L/F)=\text{Gal}(M/K)=G/H$. So $G$ would necessarily have normal subgroup $H$ and a subgroup isomorphic to $G/H.$
> 
> 
> 



Factoring a field extension into one which adds no roots of unity, followed by one which adds only roots of unity
The answer is no. The idea is to consider a tamely ramified extension of a local field. The remaining paragraphs provide details of a proof.


Take K=ℚp. Let M be a degree d unramified extension of K. Let L be an extension of M obtained by adjoining a n-th root of $pa$ for some $a$ of norm 1 in M that is not in ℚp. We will eventually take n=pd-1.


Now suppose that such a field F exists. Then F must be a totally ramified extension of ℚp with L a totally unramified extension of F. So F has degree n over ℚp. Let x be a uniformiser of F. Then $x=yb$ where $y^n=pa$ and $b$ is of norm 1. Consider the minimal polynomial of x, namely $\sum\_{i=0}^n a\_i x^i=0$ with $a\_i$ integral, WLOG $a\_n=1$. Examining valuations, $a\_0=pc$ for some integral c, and $(x^n+a\_0)/p=0 \pmod p$, ie $ab^n+c$ has zero image in the residue field.


Now to get a contracition, set n=pd-1. Thus we get $a+c$ has zero image in the finite field GF(pd) but $c \in\mathbb Q\_p$ and $a \notin \mathbb Q\_p$.


