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