Binomial theorem fun explained

The situation I described in Binomial theorem fun is also known as the Freshman’s Dream Theorem. Actually, if you consider

(a+b)^p

where a and b are members of a commutative ring of characteristic p (p prime), then

(a+b)^p=a^p+b^p

Why? The binomial theorem states that:

(a+b)^p=\sum_{k=0}^p{\binom p k a^{p-k}\cdot b^k}

Fans of number theory know that if p is prime, p divides all binomial coefficients for k between and including 1 and p-1. Thus, “mod p”, all but the first and the last parts of the above sum vanish, hence the result is (a+b)^p=a^p+b^p. Thie is also called the Frobenius Endomorphism.

In case you’re not dead yet (hehe) you might also be interested in Sophomore’s Dream, just in case you like to fill your brain with unusual stuff.

Sunday, July 19th, 2009 Math

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