I'm wondering definition of "E is a splitting over F"

Question

I know the definition of splitting field $E$ for $f(x)\in F[x]$ but I am confused by this definition: $E$ is a spliiting over $F$ (not $f(x)$).

I want to know that... please answer!

In my phone, mathematical symbol is not supported ;( I'm sorry this point. — hobin, Mar 09 '13 at 09:29

score 2 · Answer 1 · answered Mar 09 '13 at 08:50

2

Definition: Let $F$ be a field with algebraic closure $\bar{F}$. Let $$A=\{f_i(x)\mid i\in I\}$$ be a collection of polynomial in $F[x]$. A field $E\leq\bar{F}$ is the splitting field of $A$ over $F$ if $E$ is the smallest subfield of $\bar{F}$ containing $F$ and all the zeroes in $\bar{F}$ of each of the $f_i(x)$ for $i\in I$. A field $K\leq\bar{F}$ is called an splitting field over $F$ if it is the splitting field of some set of polynomials in $F[x]$

answered Mar 09 '13 at 08:50

Mikasa

67,374

Thanks :) now I have no books inevitably...;( ...... ummm then i think R is not a splitting field over Q and C is a splitting field over R, is it correct? – hobin Mar 09 '13 at 09:09
@hobin: see this link http://math.stackexchange.com/q/127110/8581 – Mikasa Mar 09 '13 at 09:32
Thanks!! I appreciate your help :) – hobin Mar 09 '13 at 09:39
Yes, thanks, good help! + – amWhy Mar 10 '13 at 04:58
@amWhy: Thanks for your kind words. I mean all of them. Thanks for your actions to me, Amy. :-) – Mikasa Mar 10 '13 at 12:00

I'm wondering definition of "E is a splitting over F"

1 Answers1