Remarks

1. The first identity can also be written as p · a = 0. In mathematical jargon, it is referred to by syaing that the characteristic of S is p.
2. The second identity is also called the Freshman's dream, as it concurs with the outcome of ordinary power expansions by many freshmen who forget about cross products.
3. The third identity is just Fermat's little theorem for finite fields! (Note that the proof does not use the particular construction of the field S.)