Exercise 6.2.12 (Number of extensions)

Proof. The number $s$ of distinct roots ${\alpha }_j^{\prime }$ of the irreducible polynomial $m$ (or ${\psi }_{\text{∗}}m$) may be less than $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(m)$ if $m$ has multiple roots.

To strengthen the hypotheses, we can assume that every irreducible polynomial $m\in K[t]$ has only simple roots in $M$ (see the notion of separable extensions in §7.2). □