Existence of splitting field

Question

Consider this theorem on existence of splitting field; Let $F$ be a field and let $f(x)$ be a non-constant element of $F[x].$ Then there exists a splitting field $E$ for $f(x)$ over $F$. In proving this theorem we use induction step on deg of $f(x)$ . If deg$f(x)$=1, then $f(x)$ is linear. so how when $f(x)$ is linear this theorem is true.

Answer 1

Hint: Prove that if $f\in F[x]$ is linear, then the splitting field of $f$ is $F$ itself.