Reference Request: Witt Vectors

Question

I'm interested in a nice (and possibly gentle) introduction to Witt vectors. It doesn't need to go into too much detail, for now, I'd just like to roughly know what they are about. So essentially my wish is something along the following lines: a definition/construction along with some motivation and possibly some suggestive examples/applications. (Though I would fancy references that do not satisfy all of this points.)

The Wikipedia article seems decent but I'm interested to know where members from MSE first learned about Witt vectors and what you would recommend.

(I was surprised by the way that there was no reference request about Witt vectors on MSE or MO. Usually, those forums would be the first place for me to check when I am looking for good book suggestions.)

Answer 1

I learned about Witt vectors from P. Schneider's lecture notes (chapter I, section 5) which, however, are in German. They are a streamlined version of the treatment by Bourbaki, which is available in English: That is §1 of book IX in Bourbaki's Commutative Algebra.

The German wikipedia article also contains an interesting motivation of the Witt vector machinery, at least for the basic case where the ring $k$ of which we take the Witt vectors is perfect. I paraphrased that in this answer on MathOverflow some time ago: https://mathoverflow.net/a/113943/27465

Answer 2

I learned about Witt vectors from Hasse's book "Zahlentheorie". There is an English translation by H.-G. Zimmer available: "Number Theory"

Answer 3

Compared to other answers I have first learned Witt vectors from a pretty unusual resource, namely Rabinoff's introductory paper, which I like mainly for the motivation section at the start. There is also an errata for it.

Answer 4

For completeness, another reference that I know is Bosch's Algebra, Chapter 4.10. (It is written in German and a classic for general/introductory algebra for German students.)

I haven't read the Witt vectors section yet, but I have invested a lot of time in other parts of the book and these were usually splendid, so I assume the Witt vectors section is also good.