Let $ f(x) ∈ F[x] \ $ and $ \ a ∈ F \ $. Show that $ \ f(x) \ $ and $ \ f(x + a) \ $ have the same splitting field over $ \ F $

Question

Let $ f(x) ∈ F[x] \ $ and $ \ a ∈ F \ $. Show that $ \ f(x) \ $ and $ \ f(x + a) \ $ have the same splitting field over $ \ F $

Answer:

Let $ \ E \ $ be the splitting field of $ \ f \ $ over $ \ F \ $.

Since $ \ E \ $ is the splitting field of $ \ f \ $ over $ \ F \ $ , $ \ E \ $ is generated over $ \ F \ $ by the roots of $ \ f \ $.

Thus $ f(x) \ $ and $ \ f(x+a ) \ $ have same splitting field $ \ E \ $ as $ \ a \in F \ $

Am I right so far ?

Help me out

Answer 1

If $x$ is a root of $f$, $f(x)=0$, this is equivalent to saying that $f(x-a+a)=0$, i.e $x-a$ is a root of $f(x+a)$, so if $x_1,..,x_n$ are the roots of $f$, $x_1-a,..,x_n-a$ are the roots of $f(x+a)$.

The field generated by $x_1,...,x_n$ contains $x_i-a$ since $a\in F$. The field generated by $x_1-a,..., x_n-a$ contains $x_i=x_i-a+a$ since $a\in F$.