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Interpretation of Abelian Extension?
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My question is if I am understanding the definition of Abelian Extension correctly because the book I am reviewing is not explicit in its definition. For some finite extension of a field $F$ by $omega$, the text says that an extension $F(omega)$ is abelian if every automorphism fixing $F$ is abelian on composition with another automorphism fixing $F$.
I am interpreting this to mean that the Galois group $Gal(F(omega):F)$ is abelian. To I have the correct idea, or am I missing an important detail?
Found an explicit definition from Wikipedia
https://en.wikipedia.org/wiki/Abelian
galois-theory galois-extensions
asked Jul 14 at 21:44
Andrew Shedlock
1106
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My question is if I am understanding the definition of Abelian Extension correctly because the book I am reviewing is not explicit in its definition. For some finite extension of a field $F$ by $omega$, the text says that an extension $F(omega)$ is abelian if every automorphism fixing $F$ is abelian on composition with another automorphism fixing $F$.
I am interpreting this to mean that the Galois group $Gal(F(omega):F)$ is abelian. To I have the correct idea, or am I missing an important detail?
Found an explicit definition from Wikipedia
https://en.wikipedia.org/wiki/Abelian
galois-theory galois-extensions
asked Jul 14 at 21:44
Andrew Shedlock
1106
I think that it is usually implied that an abelian extension is automatically also Galois. Equivalently $|Gal(F(omega):F)|=[F(omega):F]$. Otherwise, for example, $BbbQ(root3of2)/BbbQ$ would qualify simply because the group of automorphisms is trivial. Caveat: there is some variation in the definitions and notations between books. Some authors only use the notation $Gal(L:K)$ when $L/K$ is a Galois extension. Others use it to simply denote the group of automorphisms.
– Jyrki Lahtonen
Jul 14 at 21:49
@JyrkiLahtonen As in abelian extension always applies to the Galois group?
– Andrew Shedlock
Jul 14 at 21:55
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
My question is if I am understanding the definition of Abelian Extension correctly because the book I am reviewing is not explicit in its definition. For some finite extension of a field $F$ by $omega$, the text says that an extension $F(omega)$ is abelian if every automorphism fixing $F$ is abelian on composition with another automorphism fixing $F$.
I am interpreting this to mean that the Galois group $Gal(F(omega):F)$ is abelian. To I have the correct idea, or am I missing an important detail?
Found an explicit definition from Wikipedia
https://en.wikipedia.org/wiki/Abelian
galois-theory galois-extensions
asked Jul 14 at 21:44
Andrew Shedlock
1106
My question is if I am understanding the definition of Abelian Extension correctly because the book I am reviewing is not explicit in its definition. For some finite extension of a field $F$ by $omega$, the text says that an extension $F(omega)$ is abelian if every automorphism fixing $F$ is abelian on composition with another automorphism fixing $F$.
I am interpreting this to mean that the Galois group $Gal(F(omega):F)$ is abelian. To I have the correct idea, or am I missing an important detail?
Found an explicit definition from Wikipedia
https://en.wikipedia.org/wiki/Abelian
galois-theory galois-extensions
asked Jul 14 at 21:44
Andrew Shedlock
1106
edited Jul 14 at 22:01
asked Jul 14 at 21:44
Andrew Shedlock
1106
asked Jul 14 at 21:44
Andrew Shedlock
1106
asked Jul 14 at 21:44
Andrew Shedlock
1106
1106
I think that it is usually implied that an abelian extension is automatically also Galois. Equivalently $|Gal(F(omega):F)|=[F(omega):F]$. Otherwise, for example, $BbbQ(root3of2)/BbbQ$ would qualify simply because the group of automorphisms is trivial. Caveat: there is some variation in the definitions and notations between books. Some authors only use the notation $Gal(L:K)$ when $L/K$ is a Galois extension. Others use it to simply denote the group of automorphisms.
– Jyrki Lahtonen
Jul 14 at 21:49
@JyrkiLahtonen As in abelian extension always applies to the Galois group?
– Andrew Shedlock
Jul 14 at 21:55
add a comment |Â
I think that it is usually implied that an abelian extension is automatically also Galois. Equivalently $|Gal(F(omega):F)|=[F(omega):F]$. Otherwise, for example, $BbbQ(root3of2)/BbbQ$ would qualify simply because the group of automorphisms is trivial. Caveat: there is some variation in the definitions and notations between books. Some authors only use the notation $Gal(L:K)$ when $L/K$ is a Galois extension. Others use it to simply denote the group of automorphisms.
– Jyrki Lahtonen
Jul 14 at 21:49
@JyrkiLahtonen As in abelian extension always applies to the Galois group?
– Andrew Shedlock
Jul 14 at 21:55
I think that it is usually implied that an abelian extension is automatically also Galois. Equivalently $|Gal(F(omega):F)|=[F(omega):F]$. Otherwise, for example, $BbbQ(root3of2)/BbbQ$ would qualify simply because the group of automorphisms is trivial. Caveat: there is some variation in the definitions and notations between books. Some authors only use the notation $Gal(L:K)$ when $L/K$ is a Galois extension. Others use it to simply denote the group of automorphisms.
– Jyrki Lahtonen
Jul 14 at 21:49
I think that it is usually implied that an abelian extension is automatically also Galois. Equivalently $|Gal(F(omega):F)|=[F(omega):F]$. Otherwise, for example, $BbbQ(root3of2)/BbbQ$ would qualify simply because the group of automorphisms is trivial. Caveat: there is some variation in the definitions and notations between books. Some authors only use the notation $Gal(L:K)$ when $L/K$ is a Galois extension. Others use it to simply denote the group of automorphisms.
– Jyrki Lahtonen
Jul 14 at 21:49
@JyrkiLahtonen As in abelian extension always applies to the Galois group?
– Andrew Shedlock
Jul 14 at 21:55
@JyrkiLahtonen As in abelian extension always applies to the Galois group?
– Andrew Shedlock
Jul 14 at 21:55
add a comment |Â
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I think that it is usually implied that an abelian extension is automatically also Galois. Equivalently $|Gal(F(omega):F)|=[F(omega):F]$. Otherwise, for example, $BbbQ(root3of2)/BbbQ$ would qualify simply because the group of automorphisms is trivial. Caveat: there is some variation in the definitions and notations between books. Some authors only use the notation $Gal(L:K)$ when $L/K$ is a Galois extension. Others use it to simply denote the group of automorphisms.
– Jyrki Lahtonen
Jul 14 at 21:49
@JyrkiLahtonen As in abelian extension always applies to the Galois group?
– Andrew Shedlock
Jul 14 at 21:55