I've seen many times statements like:
Let K be the splitting field of P(x) over F, let G be the subfield consisting of all the roots of P(x)...
When is K different from G? IMO the splitting field is already the smallest field so there shouldn't be a proper subfield?
UPDATE
one such example can be seen here: https://link.springer.com/chapter/10.1007/978-0-387-74725-5_27?noAccess=true
On the proof part of Theorem 3, just scroll down a bit and you'll see.