p adic numbers part 3: the p-adic gamma function

This is the third part of a 3-part talk on p-adic numbers for advanced high school students. It is part of a series organized by the Berkeley mathematics circle.

We explain how to define a sort of p-adic analog of the factorial function by modifying the usualy definition of the factorial function. This gives a sort of factorial modulo any prime power, and these can be joined together to form a p-adic factorial. The construction uses Wilson's theorem, which we explain.

Links related to the video:

Further reading:
Borevich and Shafarevich, Number theory, chapter 1.3 (advanced)
J.-P. Serre, A course in arithmetic, chapter II (more advanced)
N. Koblitz, p-adic numbers, p-adic functions, and zeta functions (very advanced, for anyone who is really ambitious)

[marcfreydefont]

Great content, as usual, Sir. A pleasure to watch your courses! Keep it up.

[arturjorge]

that zeta function mic drop, what an ending

[jonka]

To feel better about the off-by-one issue, move one t to the differential and notice that dt/t is Haar measure for the multiplicative group of real numbers :)

[charlievane]

Is there any mathematical system in which there are more than one even primes ?

[theflaggeddragon]

Fantastic! Will you do lectures on algebraic number theory and possibly p adic Galois representations?

[TheAcer]

We never learned what the p-adic gamma function was :'(

[samiaario]

9:09

Imagine if it turned out that all this time mathematicians had been thinking about signs all wrong ;)