PlanetMath: extensions without unramified subextensions and class number divisibility

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extensions without unramified subextensions and class number divisibility

(Theorem)

Theorem 1 Let be an extension of number fields such that for any intermediate Galois extension , with $K\subsetneq L \subsetneq F$ , there is at least one finite place or infinite place which ramifies in the extension . Then, , the class number of , divides the class number of , .

First, we deduce some immediate corollaries.

Corollary 1 Let be an extension of number fields which is totally ramified at some prime (or at an archimedean place). Then divides .

Proof. The proof is clear since there cannot be unramified subextensions. The theorem applies. $\qedsymbol$

Corollary 2 Let be a Galois extension of number fields such that $\operatorname{Gal}(F/K)$ is a non-abelian simple group. Then divides .

Proof. In this case, there cannot be subextensions with abelian Galois group and the theorem applies. $\qedsymbol$

Proof. [Proof of the Theorem] Let

be the Hilbert class field of

. By definition,

is the maximal unramified abelian extension of

, $\operatorname{Gal}(H/K)$ is isomorphic to $\operatorname{Cl}(K)$ , the ideal class group of

and

. Since there are no nontrivial unramified abelian subextensions of

, we have $F\cap H=K$ and so

. One can show that the extension

is unramified and abelian (in fact $\operatorname{Gal}(FH/F)\cong \operatorname{Gal}(H/K)$ ). Therefore

is contained in

, the Hilbert class field of

. Hence:

$\displaystyle h_F=[L:F]=[L:FH]\cdot[FH:F]=[L:FH]\cdot [H:K]=[L:FH]\cdot h_K$

and so,

divides

. $\qedsymbol$

"extensions without unramified subextensions and class number divisibility" is owned by alozano.

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Keywords:

divisibility, class number, tower of number fields

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Cross-references: contained, ideal class group, isomorphic, abelian extension, Hilbert class field, Galois group, abelian, simple group, clear, archimedean place, prime, totally ramified, divides, class number, ramifies, finite place, Galois extension, number fields, extension
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This is version 1 of extensions without unramified subextensions and class number divisibility, born on 2005-03-10.
Object id is 6866, canonical name is ExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivisibility.
Accessed 1171 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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