Essential $p$-dimension of simple algebras and noncrossed products of prime exponent

Let $p$ be a prime and $F$ a field of characteristic different from $p$. For integers $n\ge2$ and $1\le m\le n$, we complete the determination of the essential $p$-dimension of the class of central simple algebras of degree $p^n$ and exponent dividing $p^m$. In particular, we have $\operatorname{ed}_p(\operatorname{PGL}(p^n))=p^{2n-2}+1$, extending a result of Merkurjev for $n=2$. Using the comparison theorem of Chernousov and Merkurjev, we also determine the essential $p$-dimension of every split simple group of type $A$. To obtain the lower bounds, we apply a theorem of Ofek to block centralizers. As an application, we prove the existence of division algebras of exponent $p$ over field extensions of $F$ which remain noncrossed products after any field extension of degree prime to $p$, and answer in the negative a problem in the survey by Auel, Brussel, Garibaldi, and Vishne. The algebras can be chosen to have degree $p^n$ for every $n\ge3$ if $p$ is odd and every $n\ge4$ if $p=2$. For $p=2$ and every $n\ge3$, we also determine the essential $2$-dimensions of the classes of central simple algebras of degree $2^n$ with orthogonal involution and with symplectic involution.