TITLE: revisiting $THH(\mathbb{F}_p)$
QUESTION [19 upvotes]: Reading through Bhatt-Morrow-Scholze's "Topological Hochschild Homology and Integral p-adic Hodge Theory" I encountered the following statement.

We use only “formal” properties of THH throughout the paper, with the
  one exception of Bökstedt’s computation of $THH(\mathbb{F}_p)$ ...

As a reminder, Bökstedt (and Breen?) computed $\pi_{*}THH(\mathbb{F}_p)$ as $\mathbb{F_p}[\sigma]$, where $\sigma$ is a polynomial generator in degree 2. Whereas ordinary (derived) Hochschild homology of $\mathbb{F_p}$ is easily seen to be a divided power algebra generated in the same degree, $\mathbb{F}_p\langle x\rangle.$ Note that in the latter algebra, writing $x^p = p! \frac{x^p}{p!}$ gives $x^p = 0,$ quite a different flavor than the topological theory.
At some point I looked this up and the computation appeared to rest on some not particularly formal spectral sequence manipulations which I was unable to follow. Is this still the situation? Has anyone revisited these computations recently?

REPLY [10 votes]: This computation was given a totally different proof, not using any of the ideas being discussed above by Franjou, Lannes, Schwartz in Autour de la cohomologie de Mac Lane des corps finis. [On the Mac Lane cohomology of finite fields] Invent. Math. 115 (1994), no. 3, 513–538.  The relevant groups are Tor groups in an appropriate functor category, and [FLS] use some clever resolutions, a fun but easy to prove vanishing result, and ordinary homological diagram chasing/shifting to get their result.  
All of this is much easier than the arguments by Breen and Bockstedt (or processed using Hopkins--Mahowald).  And their ideas opened the floodgates to other similar calculations -- especially after the Annals paper of Friedlander and Suslin that introduced the category of strict polynomial functors.