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the compositum of a Galois extension and another extension is Galois
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(Theorem)
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Theorem 1 Let $E/K$ be a Galois extension of fields, let $F/K$ be an arbitrary extension and assume that $E$ and $F$ are both subfields of some other larger field $T$ The compositum of $E$ and $F$ is here denoted by $EF$ Then:
- $EF$ is a Galois extension of $F$ and $E$ is Galois over $E\cap F$
- Let $H=\Gal(EF/F)$ The restriction map: \begin{eqnarray*} H=\Gal(EF/F) & \longrightarrow & \Gal(E/E\cap F)\\ \sigma & \longrightarrow & \sigma |_{E} \end{eqnarray*}is an isomorphism, where $\sigma |_{E}$ denotes the restriction of $\sigma$ to $E$
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"the compositum of a Galois extension and another extension is Galois" is owned by alozano.
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Cross-references: counterexample, example of normal extension, isomorphism, map, restriction, compositum, subfields, extension, fields, Galois extension
There is 1 reference to this entry.
This is version 3 of the compositum of a Galois extension and another extension is Galois, born on 2005-02-21, modified 2005-03-10.
Object id is 6791, canonical name is CompositumOfAGaloisExtensionAndAnotherExtensionIsGalois.
Accessed 1793 times total.
Classification:
| AMS MSC: | 12F99 (Field theory and polynomials :: Field extensions :: Miscellaneous) | | | 11R32 (Number theory :: Algebraic number theory: global fields :: Galois theory) |
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Pending Errata and Addenda
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