Let $F$ be a field and $E$ a split field of irred poly $f(x)\in F[x]$. Show that if $c,d\in F$ and $c\neq 0$ then poly $f(cx+d)$ splits in $E[x]$.

Question

Let $F$ be a field and $E$ a splitting field of irreducible polynomial $f(x)\in F[x]$. Show that if $c,d\in F$ and $c\neq 0$ then the polynomial $f(cx+d)$ splits in $E[x]$.

Let $E$ be a splitting field of $f$. If $a_1,\dots a_n$ are the roots of $f$ then $E=F(a_1,\dots,a_n)$. So I was to try and write the roots of $f(cx+d)$, $b_i$, as something that I can say are in $E$, but I am not sure how to progress. Any help is greatly appreciated! Thank you.

Answer 1

If $h\in E[x]$ is a polynomial, then the map $\nu_h:E[x]\to E[x]$, $\nu_h(g)=g\circ h$ is a ring-homomorphism. Specifically, $$\begin{align}f(cx+d)&=\nu_{cx+d}(f)\\&=LC(f)\prod_{j=1}^n (cx+d-a_j)\\&=c^nLC(f)\prod_{j=1}^n \left(x-\frac{a_j-d}c\right)\end{align}$$