Cyclotomic extensions with split Galois group - MathOverflow

Cyclotomic extensions with split Galois group

2010-11-25T00:11:40Z

$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q}$

Consider the set of all Galois extensions $E/\Q(\zeta_n)$ of a given cyclotomic field $\Q(\zeta_n)$ such that $$ \Gal(E/\Q) \simeq\Gal(E/\Q(\zeta_n)) \rtimes \Gal(\Q(\zeta_n)/\Q). $$
In other words, such that there is a homomorphism $$ \Gal(E/\Q) \leftarrow \Gal(\Q(\zeta_n)/\Q) $$ inverting the natural quotient map $$ \Gal(E/\Q) \to \frac{\Gal(E/\Q) }{\Gal(E/\Q(\zeta_n))}\simeq \Gal(\Q(\zeta_n)/\Q). $$

Are they classified? Is there a "largest" one? What can be said about them (or about their cohomology) in general? Are there any prominent examples of such extensions arising "in nature"?

Answer by Chandan Singh Dalawat for Cyclotomic extensions with split Galois group

2010-11-25T03:56:03Z

$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q} \newcommand{\Z}{\mathbf Z} \newcommand{\F}{\mathbf F}$

Abbreviate $K=\Q(\zeta_n)$. Note first that a galoisian extension $E$ of $K$ need not be galoisian over $\Q$, so I'm assuming that you are considering only those $E$ which are. We then have an exact sequenece $$ 1\to\Gal(E|K)\to\Gal(E|\Q)\to\Gal(K|\Q)\to1 $$ in which the last group is $(\Z/n\Z)^\times$, of order $\varphi(n)$. A sufficient condition for the sequence to split is : the degree $[E:K]$ is prime to $\varphi(n)$ (Schur-Zassenhaus). I don't think there is a classification of all such extensions.

Note finally that this answer does not depend on the fact that $K$ is the cyclotomic field of level $n$, or even the fact that the base field is $\Q$. It applies to any galoisian tower $E|K|F$: the associated short exact sequence $$ 1\to\Gal(E|K)\to\Gal(E|F)\to\Gal(K|F)\to1 $$ splits if the degrees $[E:K]$, $[K:F]$ are mutually prime.

Addendum (at Alex Bartel's suggestion): Let's return to the case $F=\Q$, $K=\Q(\zeta_n)$, $\Delta=\Gal(K|\Q)$, and suppose that $n$ is a prime $p$, for simplicity. Kummer theory tells us that abelian extensions $E|K$ of exponent dividing $p$ correspond bijectively to subgroups $D\subset K^\times/K^{\times p}$ under $E=K(\root p\of D)$; such an $E$ is galoisian over $\Q$ if and only if the subgroup $D$ is $\Delta$-stable. When such is the case, we get examples of the kind of extensions envisaged in the question, with "split Galois group". I guess the group $\Gal(E|\Q)$ will be commutative if and only if the $\Delta$-action on the $\F_p$-space $D$ is via the "mod $p$" cyclotomic character, namely the canonical isomorphism $\Delta\to\F_p^\times$.

Answer by Franz Lemmermeyer for Cyclotomic extensions with split Galois group

2010-11-25T09:35:37Z

The splitting of the Galois group of Hilbert class fields of an extension field is discussed in the following articles

B. Wyman, Hilbert class fields and group extensions, Scripta math. 29 (1973), 141–149
R. Gold, Hilbert class fields and split extensions, Ill. J. Math. 21 (1977), 66–69
R. Bond, On the splitting of the Hilbert class field, J. Number Theory 42 (1992), 349–360