Cyclotomic extensions with split Galois group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:27:51Z http://mathoverflow.net/feeds/question/47286 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47286/cyclotomic-extensions-with-split-galois-group Cyclotomic extensions with split Galois group Jon Yard 2010-11-25T00:11:40Z 2010-11-27T09:19:14Z <p>$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q}$</p> <p>Consider the set of all Galois extensions $E/\Q(\zeta_n)$ of a given cyclotomic field $\Q(\zeta_n)$ such that $$\Gal(E/\Q) \simeq\Gal(E/\Q(\zeta_n)) \rtimes \Gal(\Q(\zeta_n)/\Q).$$<br> In other words, such that there is a homomorphism $$\Gal(E/\Q) \leftarrow \Gal(\Q(\zeta_n)/\Q)$$ inverting the natural quotient map $$\Gal(E/\Q) \to \frac{\Gal(E/\Q) }{\Gal(E/\Q(\zeta_n))}\simeq \Gal(\Q(\zeta_n)/\Q).$$</p> <p>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"?</p> http://mathoverflow.net/questions/47286/cyclotomic-extensions-with-split-galois-group/47297#47297 Answer by Chandan Singh Dalawat for Cyclotomic extensions with split Galois group Chandan Singh Dalawat 2010-11-25T03:56:03Z 2010-11-27T09:19:14Z <p>$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q} \newcommand{\Z}{\mathbf Z} \newcommand{\F}{\mathbf F}$</p> <p>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.</p> <p>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.</p> <p><strong>Addendum</strong> (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 <em>prime</em> $p$, 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 $\Delta\to\F_p^\times$. </p> http://mathoverflow.net/questions/47286/cyclotomic-extensions-with-split-galois-group/47316#47316 Answer by Franz Lemmermeyer for Cyclotomic extensions with split Galois group Franz Lemmermeyer 2010-11-25T09:35:37Z 2010-11-25T09:35:37Z <p>The splitting of the Galois group of Hilbert class fields of an extension field is discussed in the following articles</p> <ul> <li>B. Wyman, Hilbert class fields and group extensions, Scripta math. 29 (1973), 141–149</li> <li>R. Gold, Hilbert class fields and split extensions, Ill. J. Math. 21 (1977), 66–69</li> <li>R. Bond, On the splitting of the Hilbert class field, J. Number Theory 42 (1992), 349–360</li> </ul>