The situation for this question deals with the varying definitions of the Galois group of a field extension.
The question is how Dummit and Foote defines the Galois group. I'll list their definition after the specific example set up. In my Algebra class if $K/F$ if a field extension we define $Gal(K/F) = \{\phi \in Aut(K): \phi(f) = f,\,\, \forall f\in F \}$. i.e. the automorphisms of $K$ that fix $F$. We were asked to find the Galois group of the splitting field of $x^3 - 10$ over $\mathbb{Q}$.
The roots of this polynomial are $\sqrt[3]{10}\zeta_i $ where $\zeta_i$ are the are the 3rd roots of unity. Thus the splitting field for this polynomial is $\mathbb{Q}(\sqrt[3]{10},\zeta_i)$. The degree of this extension ($|\mathbb{Q}(\sqrt[3]{10},\zeta_i):\mathbb{Q}|$) is 9, as can be seen by extending $\mathbb{Q}$ one root at a time.
$$|\mathbb{Q}(\sqrt[3]{10},\zeta_i):\mathbb{Q}| = |\mathbb{Q}(\sqrt[3]{10},\zeta_i):\mathbb{Q}(\sqrt[3]{10})|\cdot |\mathbb{Q}(\sqrt[3]{10}):\mathbb{Q}|$$
where each extension is degree 3. And after some work I have determined that $Aut(K/F) = S_3$ (i.e. from class definition the Galois group is $S_3$).
Now on to the main question: According to Dummit and Foote's definition a group is Galois if $|Aut(K/F)| = |K:F|$ so in our example $|Aut(\mathbb{Q}(\sqrt[3]{10},\zeta_i)| = 6 \not= 9 = |\mathbb{Q}(\sqrt[3]{10},\zeta_i):\mathbb{Q}|$. Thus by this definition our group is not a Galois group.
The very next Corollary [Cor 14.6] states: If $K$ is the splitting field over $F$ of a separable polynomial $f(x)$ then $K/F$ is Galois.
Consider the polynomial $p(x) = x^3 - 10$ again. The splitting field is the one listed above $\mathbb{Q}(\sqrt[3]{10},\zeta_i)$ according to Dummit and Foote $p(x)$ is separable since no root has multiplicity greater than 1 (they give an example where $x^2 - 2$ is separable over $\mathbb{Q}$ for similar reasonings). Thus by their definition $Aut(\mathbb{Q}(\sqrt[3]{10},\zeta_i))$ is not the Galois group but by their Corollary it is, which is very confusing.
Any clarity? Any mistakes computing degrees or Automorphism groups?
A few other optional things of note:
(i) If there has been an error in saying the splitting field of $p$ is $\mathbb{Q}(\sqrt[3]{10},\zeta_i)$ for some reason, it certainly is the splitting field of $q(x) = (x^3 - 10)(x^3 -1)$ which by their definition is still separable and seems to still break their definition.
(ii) A different book (Basic Algebra by Knapp) defines separable as being able to be split into distinct linear factors, with this definition of separable using Dummit and Foote's Corollary keeps us consistent, but it seems odd that Dummit and Foote's definitions could contradict each other.
Unfortunately I can't seem to make this any shorter, it all seems really important to the question. Any help would be appreciated, Thanks.
EDIT: Okay guys disregard this question. I was mistaken in the minpoly of $\zeta_3$ over $\mathbb{Q}$ This is a degree 6 extension.
