How is this induction argument here valid

Elaborating on Lee Mosher's comment, consider the following assertion, $P(n)$, about a natural number $n$:

$P(n)$: Let $F$ be a field and let $f(x)$ be a non-constant element of $F[x]$ of degree at most $n$. Then there exists a splitting field $E$ for $f(x)$ over $F$.

The statement you want to prove can be written as $(\forall n)(P(n))$ and is thus a candidate for a proof by induction.

The fact that $P(n)$ itself contains two implicit universal quantifiers, including "for all fields $F$", might seem too good to be true ... but mathematics is, quite simply, that good! especially (as we see here) for its ability to package whole infinite families of assertions together in a single statement.

The full statement of the induction hypothesis is

The theorem is true for all fields $F$ and for all non-constant elements of $F[x]$ having degree $\le n$.

You'll find that the induction hypothesis is often not spelled out in full formal detail like that. Instead it becomes the reader's responsibility to figure out the appropriate formal statement of the induction hypothesis.