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In Artin's lecture notes on Galois Theory (Dover version), Theorem 15 page 44 says that "a field $E$ is a normal extension of field $F$ if and only if $E$ is the splitting field of a separable polynomial $p(x)$ in $F$." Since every algebraic extension generated by a finite number of separable elements is separable, this implies that every normal extension is separable.

On the other hand, the wikipedia page on "Galois extension" defines the Galois extension to be an algebraic extension which is both normal and separable.

It seems that the requirement that the extension is both normal and separable is redundant, since normal does imply separable. Any comments?

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  • $\begingroup$ Artin's definition is no longer standard, and is not equivalent to the wikipedia definition of normal extension $\endgroup$ – Cocopuffs Aug 26 '13 at 21:43
  • $\begingroup$ @Cocopuffs: I see...thanks. $\endgroup$ – Manos Aug 26 '13 at 21:45
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The definition of "normal" that Artin is using is non-standard. From p.41 of Artin's notes:

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The standard definition is given in Lang's Algebra, p.237-238:

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(proof of Theorem 3.3)

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Using the standard definition, a normal field extension need not be separable, e.g. the extension $\mathbb{F}_p(T^{1/p})/\mathbb{F}_p(T)$, nor must it be of finite degree, e.g. the extension $\overline{\mathbb{Q}}/\mathbb{Q}$.

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  • $\begingroup$ I see. So what Artin calls normal, now is called Galois. Coorect? $\endgroup$ – Manos Aug 26 '13 at 21:48
  • $\begingroup$ @Manos: It would now be called a finite Galois extension; Artin's definition forces the degree of the extension to be finite. In contrast, the modern definition of Galois (normal + separable) allows for infinite Galois extensions. $\endgroup$ – Zev Chonoles Aug 26 '13 at 21:51

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