issam el mariami
Thread moved from the technical forums to the schoolwork forums.
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]
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]
Last edited by a moderator: