# Tag Archives: Number Theory

## Euler Phi-Function is a Multiplicative Function

Let $$n$$ be a positive integer. Recall that the Euler phi-function $$\phi(n)$$ is defined as the number of positive integers less than or equal to $$n$$ and relatively prime to $$n.$$ Note that $$\phi(1)=1.$$ We have seen that Euler used this function to generalize the Fermat's Little Theorem. It is sometimes needed to calculate the value $$\phi(n)$$ of… Read More »

## Euler’s Generalization of Fermat’s Little Theorem

Fermat's Little Theorem says: Theorem 1. (Fermat) If $$p$$ is a prime number and $$(a,~p)=1,$$ that is, if $$a$$ and $$p$$ are relatively primes, then $$a^{p-1}\equiv 1$$ $$({\rm mod}~ p).$$ Euler gave a generalization of Fermat's theorem. His generalization will follow at once from next theorem, which is proceed by counting, using essentially the same argument as in… Read More »

## Fermat’s Little Theorem and Finite Prime Fields

The fact that "$$\mathbb{Z}_p$$ is a field if and only if $$p$$ is a prime" can be derived from the fact that "every finite integral domain is a field." Also, it is easily derived that every field include a subfield which is isomorphic to one of $$\mathbb{Q}$$ or $$\mathbb{Z}_p$$ for a prime $$p$$. That's why the fields $$\mathbb{Z}_p$$… Read More »