Confusion about definition of a splitting field

issam el mariami
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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]
 
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##(x^2+1)^2## isn't irreducible and therefore not separable.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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