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## Saturday, March 3, 2012

### An algebraic proof of Fermat's Little Theorem

Let $G$ be an abelian group. Define a scalar multiplication over $\mathbb{Z}$ as follows: $$n \cdot g = \underbrace{g+g+\cdots+g}_{n \ \text{times}}.$$ Note that in this case $|G| \ g=0$. ( We turned $G$ into a $\mathbb{Z}$-module. )

For primes, the multiplicative group $\mathbb{Z}_p$ is abelian, $| \mathbb{Z}_p | = p-1$ and the identity element is $1$. Let $a \in \mathbb{Z}_p$ and the multiplicative notation of $|G| \ g=0$ becomes $a^{p-1} \equiv 1 \bmod{p}$. But this is just Fermat's Little Theorem!

 Fermat

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)