Background for Gaussian Integers?

[sutupidmath]

Hi,

I was wondering if having some training in algebraic number theory is a must for even starting to work with Gaussian Integers, or one can work with them with some knowledge of abstract algebra, like group, ring and field theory knowledge (i.e. 1 year of undergraduate abstract algebra)?


Also does anyone know any book that treats Gaussian integers in some more depth?

Thanks!

 

[micromass]

It depends on what you want to do with them. But I think it's very possible to understand theory about Gaussian integers without much knowledge of number theory. Some ring theory (= knowledge about rings, UFD's, PID's, Euclidean domains) is already a good starting point.

I can't immediately provide a reference, but I'll search one for you. Can you perhaps be a bit more specific in what you want to do with Gaussian integers??

 

[sutupidmath]

It depends on what you want to do with them. But I think it's very possible to understand theory about Gaussian integers without much knowledge of number theory. Some ring theory (= knowledge about rings, UFD's, PID's, Euclidean domains) is already a good starting point.

I can't immediately provide a reference, but I'll search one for you. Can you perhaps be a bit more specific in what you want to do with Gaussian integers??

Well, the short term goal is to be able to understand the theory about Gaussian integers. However on the long run, it seemed like an interesting field for research too. So, I am in the lookout for a topic for my senior research thesis/project (which will be this fall), and Gaussian integers stroke me as interesting. So, I was thinking spending some time during the summer learning the proper background, and maybe in the fall starting to think about doing research.

 

[micromass]

Here are some references containing the basics of Gaussian integers and some applications:

www.math.uconn.edu/~troby/Math3240F10/Zinotes.pdf[/URL]
[url]www.math.ou.edu/~kmartin/nti/chap6.pdf[/url]
[url]www.oberlin.edu/faculty/jcalcut/gausspi.pdf[/url]

If you understand what is in these texts, then you can probably go to more advanced stuff. The problem is that the advanced stuff will probably involve texts in number theory. So you probably need to study that next...

 

[mathwonk]

check out shifrin's algebra from a geometric viewpoint, or mike artin's algebra.