Gaussian Primes etc

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fyggy
Posts: 1
Joined: Thu May 16, 2019 10:34 pm UTC

Gaussian Primes etc

Postby fyggy » Thu May 16, 2019 10:47 pm UTC

I'm sure some people here are familiar with Gaussian Primes; where you use the Gaussian numbers to factorise other Gaussian numbers and count the factors. IIRC, if they have 8 factors, then they are a Gaussian Prime. Now, I was wondering if this would be possible with other complex number systems, like [url href = "http://forums.xkcd.com/viewtopic.php?t=101814"] the one mentioned here[/url], or the one used in quaternions. My main concern is non-commutability, but I don't really know enough about complex numbers to make any conclusions. I also don't know if this forum is still alive, but it's worth a shot. Thanks in advance???!??!

DavCrav
Posts: 251
Joined: Tue Aug 12, 2008 3:04 pm UTC
Location: Oxford, UK

Re: Gaussian Primes etc

Postby DavCrav » Sun May 19, 2019 1:33 pm UTC

What you need is a Unique Factorization Domain, or UFD, then you define primes. In fact, primes and irreducibles can always be defined. An element x is prime if whenever x divides ab, x divides a or x divides b. An element x is irreducible if, whenever x=ab, then one of a and b is a unit. (A unit is an element u such that there exists v with uv=1.) Primes are always irreducible, but the converse is not true. If it is, then the ring is called a UFD. They are precisely the rings where a version of the fundamental theorem of arithmetic holds.

The quaternions (with entries in the integers) do form a non-commutative unique factorization domain.


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