Invertible Elements in Zn and Factoring over Z[i]

[Yapper]

Homework Statement

Can someone explain the notation of the integer ring(Z)? My friend's in Math 100 and has questions on Z. I have never heard of ring's before and I'm at a loss to help him. From what I've gathered Z is a set of all integers. They ask for the invertible elements in different Zn like Z1 Zi Z101 if someone could explain how to identify invertible elements in the different Zn would be great.

They also ask him to "Factor 5 + i over Z."

The Attempt at a Solution

I've tried to look up rings, but I don't understand how you would factor a number over a set especially when it has an imaginary in there, haven't really worked with them that much. No text for the class and no explanation from the instructor.

 

[Dick]

It's nice of you to try and help your friend. But since he's taking the course maybe you should ask him to explain some of this stuff to you. Z(n) is the set of integer mod n. x has an inverse if there is a number y such that x*y=1 mod n. It has a fair amount to do with whether the gcd(x,n) is 1 or not. Z is the integers extended by the imaginary unit, i. I.e. all numbers of the form n+mi with n and m integers. Does (5+i) equal a product of two of those numbers in a nontrivial way?

 

[Yapper]

thanks, and he says his teacher didn't explain it to him at all so that's why I'm doing this for him.

If you could clarify, what does mod n mean? And how would he go about factoring 5 + i over Z? would it just be 5?

 

[HallsofIvy]

What he's really saying is that he didn't pay attention while his teacher explained it. Can he at least bother to read the textbook?

 

[Dick]

Maybe your friend could post his questions directly here or loan you his textbook so you can look these things up? If not there are other sources of definitions besides here. Try looking up 'modular arithmetic' for the first topic and 'gaussian integers' for the second.

 

[Yapper]

wow do you guys read, this is a math 100 course at a ****ty university, you automatically assume that the teacher is perfect, and there is no textbook for the 2nd time, but thanks for the topics of research dick i have a starting point