Confusion about definition of a splitting field

[issam el mariami]

TL;DR Summary: Confision about definition of a splitting field

I have an exercise that tells me to show that if $F$is a field and $f(x)$ is a polynomial over F which irreducible of degree n and $E/F$ is a splitting field of $f(x)$ then n divides $[E:F]$, DONE!

The second part is what bothers me, it says: now suppose $f(x)$ is separable, show that n divides the order of $Gal(E/F)$. See exercise 78, J Rotman - Galois Theory (Universitext), second edition. Now from the theorem 56 (of the same book) I can tell that $|Gal(E/F)|=[E:F]$, nice!!!.

Now, for sure I need to apply part one of the exercise, so here how it goes: from the definition of a separable polynomial we know that (stik right now for just the case of two factors, because the general case cane be done by induction) $f(x)= a f_1(x) f_2(x)$ where a is an element of F and the two foregoing polynomials are irreducible (NOT NECESSARILY DISTINCT) and EACH one of them has no repeated roots.

But, then It cames to my mind that $|\mathbb{C}: \mathbb{R}|=2$, and $(x^2+1)(x^2+1)$ is separable over $\mathbb{R}$ with degree four, But $x^2+1$ (itself!) is separable so by the same theorem $|Gal(\mathbb{C}:\mathbb{R})|=2$ But four does not divide two. Could someone please tell me where I messed up, and if I'm right about using induction to resolve the problem (so $deg(f_1(x))deg(f_2(x))=deg(f(x)$) divides $[E:F]$).

[LaTeX edited by a Mentor]

 

[fresh_42]

$(x^2+1)^2$ isn't irreducible and therefore not separable.