Prime Characteristic
When p is a prime number and x and y are members of a commutative ring of characteristic p, then (x + y)p = xp + yp. This can be seen by examining the prime factors of the binomial coefficients: the nth binomial coefficient is
The numerator is p factorial, which is divisible by p. However, when 0 < n < p, neither n! nor (p − n)! is divisible by p since all the terms are less than p and p is prime. Since a binomial coefficient is always an integer, the nth binomial coefficient is divisible by p and hence equal to 0 in the ring. We are left with the zeroth and pth coefficients, which both equal 1, yielding the desired equation.
Thus in characteristic p the freshman's dream is a valid identity. This result demonstrates that exponentiation by p produces an endomorphism, known as the Frobenius endomorphism of the ring.
The demand that the characteristic p be a prime number is central to the truth of the freshman's dream. In fact, a related theorem states that a number n is prime if and only if (x+1)n ≡ xn + 1 (mod n) in the polynomial ring . This theorem is a direct consequence of Fermat's Little Theorem and it is a key fact in modern primality testing.
Read more about this topic: Freshman's Dream
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