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[parent] unramified extensions and class number divisibility (Corollary)

The following is a corollary of the existence of the Hilbert class field.

Corollary 1   Let $ K$ be a number field, $ h_K$ is its class number and let $ p$ be a prime. Then $ K$ has an everywhere unramified Galois extension of degree $ p$ if and only if $ h_K$ is divisible by $ p$.
Proof. Let $ K$ be a number field and let $ H$ be the Hilbert class field of $ K$. Then:
$\displaystyle \vert\operatorname{Gal}(H/K)\vert=[H:K]=h_K.$
Let $ p$ be a prime number. Suppose that there exists a Galois extension $ F/K$, such that $ [F:K]=p$ and $ F/K$ is everywhere unramified. Notice that any Galois extension of prime degree is abelian (because any group of prime degree $ p$ is abelian, isomorphic to $ \mathbb{Z}/p\mathbb{Z}$). Since $ H$ is the maximal abelian unramified extension of $ K$ the following inclusions occur:
$\displaystyle K \subsetneq F\subseteq H$
Moreover,
$\displaystyle h_K=[H:K]=[H:F]\cdot[F:K]=[H:F]\cdot p.$
Therefore $ p$ divides $ h_K$.

Next we prove the remaining direction. Suppose that $ p$ divides $ h_K=\vert\operatorname{Gal}(H/K)\vert$. Since $ G=\operatorname{Gal}(H/K)$ is an abelian group (isomorphic to the class group of $ K$) there exists a normal subgroup $ J$ of $ G$ such that $ \vert G/J\vert=p$. Let $ F=H^J$ be the fixed field by the subgroup $ J$, which is, by the main theorem of Galois theory, a Galois extension of $ K$. This field satisfies $ [F:K]=p$ and, since $ F$ is included in $ H$, the extension $ F/K$ is abelian and everywhere unramified, as claimed. $ \qedsymbol$



"unramified extensions and class number divisibility" is owned by alozano.
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See Also: ideal class, $p$-extension, ramification index, topics on ideal class groups and discriminants

Keywords:  class number divisibility

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class number divisibility in extensions (Theorem) by alozano
extensions without unramified subextensions and class number divisibility (Theorem) by alozano
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Cross-references: field, Galois theory, subgroup, fixed field, normal subgroup, class group, abelian group, divides, inclusions, extension, isomorphic, group, abelian, prime number, divisible, degree, Galois extension, unramified, prime, class number, number field, Hilbert class field
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This is version 2 of unramified extensions and class number divisibility, born on 2005-02-17, modified 2005-02-17.
Object id is 6765, canonical name is UnramifiedExtensionsAndClassNumberDivisibility.
Accessed 1745 times total.

Classification:
AMS MSC: 11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants)
 11R32 (Number theory :: Algebraic number theory: global fields :: Galois theory)
 11R37 (Number theory :: Algebraic number theory: global fields :: Class field theory)
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