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Community Bot
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The answer to whether this is possible for general fields is no. However, the counterexamplescounterexamples used two ingredients:

  1. $\Bbb Q_p$, whose extensions $K$ containing $\Bbb Q_p(\sqrt[p^e]{u})$ might not be not Galois even if $\zeta_{p^e}\in K$.

  2. Ramified extensions.

IF one or both of these is not allowed, then will this result then hold? For example:

If $k$ is a number field, and $L/k$ is an unramified / Abelian extension, can we decompose $L = K(\zeta_n)$, where $K$ and $k$ have the same number of roots of unity?

The answer to whether this is possible for general fields is no. However, the counterexamples used two ingredients:

  1. $\Bbb Q_p$, whose extensions $K$ containing $\Bbb Q_p(\sqrt[p^e]{u})$ might not be not Galois even if $\zeta_{p^e}\in K$.

  2. Ramified extensions.

IF one or both of these is not allowed, then will this result then hold? For example:

If $k$ is a number field, and $L/k$ is an unramified / Abelian extension, can we decompose $L = K(\zeta_n)$, where $K$ and $k$ have the same number of roots of unity?

The answer to whether this is possible for general fields is no. However, the counterexamples used two ingredients:

  1. $\Bbb Q_p$, whose extensions $K$ containing $\Bbb Q_p(\sqrt[p^e]{u})$ might not be not Galois even if $\zeta_{p^e}\in K$.

  2. Ramified extensions.

IF one or both of these is not allowed, then will this result then hold? For example:

If $k$ is a number field, and $L/k$ is an unramified / Abelian extension, can we decompose $L = K(\zeta_n)$, where $K$ and $k$ have the same number of roots of unity?

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Alex
Alex
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When is possible to factor a field extension into one which adds no roots of unity, followed by one which adds only roots of unity?

The answer to whether this is possible for general fields is no. However, the counterexamples used two ingredients:

  1. $\Bbb Q_p$, whose extensions $K$ containing $\Bbb Q_p(\sqrt[p^e]{u})$ might not be not Galois even if $\zeta_{p^e}\in K$.

  2. Ramified extensions.

IF one or both of these is not allowed, then will this result then hold? For example:

If $k$ is a number field, and $L/k$ is an unramified / Abelian extension, can we decompose $L = K(\zeta_n)$, where $K$ and $k$ have the same number of roots of unity?