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[parent] class number divisibility in extensions (Theorem)

Throughout this entry we will use the following corollary to the existence of the Hilbert class field (see the parent entry, unramified extensions and class number divisibility for details of the proof).

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$.

In this entry we are concerned about the divisibility properties of class numbers of number fields in extensions.

Theorem 1   Let $ F/K$ be a Galois extension of number fields, let $ h_F$ and $ h_K$ be their respective class numbers and let $ p$ be a prime number such that $ p$ does not divide $ [F:K]$, the degree of the extension. Then, if $ p$ divides $ h_K$, the class number of $ F$, $ h_F$, is also divisible by $ p$.
Proof. Let $ F$, $ K$ and $ p$ be as in the statement of the theorem. Assume that $ p\vert h_K$. Thus, by the corollary above, there exists an unramified Galois extension field $ E$ of $ K$ of degree $ p$. Notice that the fact that $ p$ does not divide the degree of the extension $ F/K$ implies that $ F\cap E=K$. In particular, the compositum $ FE$ is a Galois extension of $ F$ and
$\displaystyle \operatorname{Gal}(FE/F)\cong \operatorname{Gal}(E/E\cap F)\cong \operatorname{Gal}(E/K).$
Thus, the extension $ FE/F$ is of degree $ p$, Galois, and therefore abelian. By the corollary above, in order to prove the theorem it suffices to show that the extension $ FE/F$ is unramified. Suppose for a contradiction that $ \mathcal{Q}_F$ is a prime ideal which ramifies in the extension $ FE/F$. Let $ \mathcal{Q}_{EF}$ be a prime lying above $ \mathcal{Q}_F$ and let $ \mathcal{Q}_K$ be a prime of $ K$ such that $ \mathcal{Q}_F$ lies above it. Similarly, let $ \mathcal{Q}_E$ be a prime of $ E$ lying above $ \mathcal{Q}_K$ and such that the prime $ \mathcal{Q}_{EF}$ lies above $ \mathcal{Q}_E$. For an arbitrary extension $ A/B$, the ramification index of a prime $ \mathcal{Q}_A$ is denoted by $ e(\mathcal{Q}_B\vert\mathcal{Q}_A)$. Then, by the multiplicativity of the ramification index in towers, we have:
$\displaystyle e(\mathcal{Q}_{EF}\vert\mathcal{Q}_K)=e(\mathcal{Q}_{EF}\vert\mat... ...e(\mathcal{Q}_{EF}\vert\mathcal{Q}_E)\cdot e(\mathcal{Q}_{E}\vert\mathcal{Q}_K)$
Since we assumed that $ \mathcal{Q}_F$ is ramified in $ EF/F$, and the degree of the extension is $ p$, we must have $ e(\mathcal{Q}_{EF}\vert\mathcal{Q}_F)=p$. Therefore, by the equality above, $ p$ divides $ e(\mathcal{Q}_{EF}\vert\mathcal{Q}_E)\cdot e(\mathcal{Q}_{E}\vert\mathcal{Q}_K)$. Notice that the extension $ E/K$ is everywhere unramified, therefore $ e(\mathcal{Q}_E\vert\mathcal{Q}_K)=1$. Also, $ [EF:E]=[F:K]$ which, by hypothesis, is relatively prime to $ p$. Thus $ e(\mathcal{Q}_{EF}\vert\mathcal{Q}_E)$ is also relatively prime to $ p$, and so, $ p$ is not a divisor of $ e(\mathcal{Q}_{EF}\vert\mathcal{Q}_E)\cdot e(\mathcal{Q}_{E}\vert\mathcal{Q}_K)$, which leads to the desired contradiction, finishing the proof of the theorem. $ \qedsymbol$

Also, read the entry extensions without unramified subextensions and class number divisibility for a similar and more general result.



"class number divisibility in extensions" is owned by alozano.
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See Also: ideal class, existence of Hilbert class field, the compositum of a Galois extension and another extension is Galois, decomposition group, extensions without unramified subextensions and class number divisibility, topics on ideal class groups and discriminants

Keywords:  divisibility, class number, tower of number fields

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Attachments:
push-down theorem on class numbers (Theorem) by alozano
class number divisibility in $p$-extensions (Theorem) by alozano
class number divisibility in cyclic extensions (Theorem) by alozano
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Cross-references: similar, extensions without unramified subextensions and class number divisibility, divisor, relatively prime, hypothesis, equality, prime ideal, contradiction, order, abelian, compositum, implies, field, prime number, extensions, properties, divisibility, divisible, degree, Galois extension, unramified, prime, class number, number field, unramified extensions and class number divisibility, parent, Hilbert class field
There are 2 references to this entry.

This is version 6 of class number divisibility in extensions, born on 2005-02-21, modified 2006-01-30.
Object id is 6792, canonical name is ClassNumberDivisibilityInExtensions.
Accessed 1173 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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