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extensions without unramified subextensions and class number divisibility
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(Theorem)
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First, we deduce some immediate corollaries.
Proof. The proof is clear since there cannot be unramified subextensions. The theorem applies. 
Corollary 2 Let $F/K$ be a Galois extension of number fields such that $\Gal(F/K)$ is a non-abelian simple group. Then $h_K$ divides $h_F$ .
Proof. In this case, there cannot be subextensions with abelian Galois group and the theorem applies. 
Proof. [Proof of the Theorem] Let $H$ be the Hilbert class field of $K$ . By definition, $H$ is the maximal unramified abelian extension of $K$ , $\Gal(H/K)$ is isomorphic to $\Cl(K)$ , the ideal class group of $K$ and $[H:K]=h_K$ . Since there are no nontrivial unramified abelian subextensions of $F/K$ , we have $F\cap H=K$ and so $[FH:F]=[H:K]=h_K$ . One can show that the extension $FH/F$ is unramified and abelian (in fact $\Gal(FH/F)\cong \Gal(H/K)$ ). Therefore $FH$ is contained in $L$ , the Hilbert class field of $F$ . Hence: $$h_F=[L:F]=[L:FH]\cdot[FH:F]=[L:FH]\cdot [H:K]=[L:FH]\cdot h_K$$ and so, $h_K$ divides $h_F$ . 
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"extensions without unramified subextensions and class number divisibility" is owned by alozano.
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Cross-references: contained, ideal class group, isomorphic, abelian extension, Hilbert class field, Galois group, abelian, simple group, non-Abelian, theorem, clear, proof, archimedean place, prime, totally ramified, divides, class number, ramifies, finite place, Galois extension, number fields, extension
There are 2 references to this entry.
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 1541 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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Pending Errata and Addenda
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