# Category Archives: Mathematik

## Every finite division ring is a field

It is well-known for undergraduate students who major in mathematics that every finite integral domain is a field. Theorem 1. Every finite integral domain is a field. Proof. Let $$0,~1,~a_1 ,~a_2 ,~ \cdots ,~ a_n$$ be all the elements of a finite integral domain $$D .$$ We need to show that for $$a \in D$$, where… Read More »

## Which is better, e or π?

Top ln(e10) reasons why e is better than pi : e is easier to spell than pi. pi ≒ 3.14 while e ≒ 2.718281828459045. The character for e can be found on a keyboard, but pi sure can't. Everybody fights for their piece of the pie. ln(pi1) is a really nasty number, but ln(e1) = 1. e is… Read More »

## 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 »

## “Associativity, Left Identity, Left Inverse Elements” Makes a Group

In most undergraduate algebra textbooks, you can find the definition of a group as following: A nonempty set $$G$$ with a binary operation satisfying the following conditions is called a group: Associative rule: $$a(bc)=(ab)c$$ for all $$a,~b,~c \in G$$. Existence of an identity element: There exists $$e\in G$$ s.t. $$ea = ae = a$$ for all \(a \in… Read More »