1. ## Complex prime numbers?

Hello!

I just want to know if there is anything like complex prime numbers, and what work has been done in the subject.

Ordinary prime numbers is just there if you want to factorise real integer numbers, not even zero or negative number is factorable, case 0 and -1 is missing prime numbers in that case. You start with the multiplicative identity, 1, so the number 1 doesn't need to have any prime numbers to be achieved. But if one want to get 0, multiplication by 0 is necessary, but zero is not a prime number. And if one want to get a negative number, multiplication by -1 is necessary, wich is not a prime number either.

But if we want every complex number with integer coordinates to be factorable, we'll have to introduce other prime numbers. 0 is one of them, i is another. Cause, to reach the negative numbers, you could multiply by -1, but to reach the imaginary numbers, one will have to multiply by i, and i*i = -1. But i can't be achieved by multiplying other prime numbers with each other, possible if you introduced -i instead, but having two different prime numbers which can be factorized by each other isn't such a good idea, and i is more natural than -i. Now every number in the 2:nd, 3:d or 4:th quadrant, can be written by a number in the first quadrant times a power to i. Every negative number or imaginary number can be written as a positive number times a power to i. Clearly the first number in the first quadrant we hit is (1, 1), so that can be made a prime number. Now, (2, 1) and (1, 2) can be made prime numbers as well. In fact, every number (a, 1) can be made a prime number. But what about the numbers (1, a), are those prime numbers? What about the rest of the numbers in the first quadrant? Is there some method to get those?

This is my conclusions: The ordinary prime numbers still exist at the possitive real axis. No imaginary prime number more than i and 0 exists, since the rest of the numbers at the imaginary axis can be written as the same real number times i if the number is possitive, and times i^3 if the number is negative. The same rule still exists that no prime number shall be able to be written as a product of other prime numbers. On the other hand, prime number i can always be written as i^5. The other way around, a big or a negative exponent of i can always be reduced to an exponent that lays in some specific interval, like [0, 4[, or maybe ]-2, 2]. i however along with zero is the only prime number which can be factorized with another set of prime factors than the set only containing the original prime number.

2. Originally Posted by TriKri
The ordinary prime numbers still exist at the positive real axis.
That is clearly true!
Every real number is a complex number.
So prime in the reals then prime in the complex.
No problem there, correct?

What do you understand a prime complex number to be?
You may want to look-up Gaussian-integers.
See if you can use that concept to forward your question.

3. There is actually a very abstract notion of what a prime number is. But I do not know if you have the background to understand it. Have you ever studied basic abstract algebra and basic field theory?

Definition 1: Let $D$ be an integral domain we say $a|b$ (reads $a$ divides $b$) iff there exists $c$ so that $b = ac$.

Definition 2: A "unit" in $D$ is an element $u$ so that $u|1$.

Definition 3: A non-unit non-zero number is called "irreducible" iff it cannot be factored as a product of two non-unit numbers.

Definition 4: A non-unit non-zero element $p$ is called "prime" iff $p|(ab)\implies p|a \mbox{ or }p|b$.

Theorem 1: If $p$ is prime then it is irreducible.

We ask is the converse true? In fact it is not!

Definition 5: An integral domain is "principal" (called "principal ideal domain") iff every ideal is principal.

Theorem: For every PID any irreducible elements are prime elements.

So as you can see there is a very abstract approach to the meaning of what "prime" is.

As an example, you can work in $\mathbb{Z}[i] = \{ a+ bi | a,b\in \mathbb{Z} \}$ called the "Gaussian integers". And look for primes.

The two great mathematicians that should be credited for this abstract approach to factorization are Kummer and his student Kronecker.