# Thread: Characteristic of a field

1. ## Characteristic of a field

let F be a field with characteristic p, a prime. show that (a+b)^p = a^p + b^p for all a,b in F.

Using pascal's triangle it's easy to show for the first few primes that this is true, yet I don't know how to prove it for all primes, p. I was thinking of using induction, but don't know exactly how to go about doing that. Any help would be appreciated.

2. Originally Posted by Coda202
let F be a field with characteristic p, a prime. show that (a+b)^p = a^p + b^p for all a,b in F.

Using pascal's triangle it's easy to show for the first few primes that this is true, yet I don't know how to prove it for all primes, p. I was thinking of using induction, but don't know exactly how to go about doing that. Any help would be appreciated.
This question has been posted on MHF before many times. There are two hints.

Hint 1: Use binomial theorem.

Hint 2: Note that $\displaystyle p$ divides $\displaystyle {p\choose j}$ for $\displaystyle 0 < j < p$.