Multiplication tables of rings

[phyguy321]

Homework Statement

construct a multiplication table for the ring Z_{3}[\alpha], \alpha^{2} + 1(bar) = 0(bar)

Homework Equations

The Attempt at a Solution

I'm actually confused on how to find the elements of the ring. My book and notes have thrown me off a bit and I can't find them. Hint: there are 9 elements

 

[jacobrhcp]

I think for advanced courses like yours it might help me and others if you remind us of what things like (bar) mean

and do you mean by Z^3 actually Z_3 or ZxZxZ or something completely different outside my knowledge?

 

[phyguy321]

yea sorry the text editor on here sucks...it is supposed to be Z_3 (which is what i put) and (bar) refers to being part of that equivalence class so 1 bar is ...-2,1,4,7... and 0 bar is ...-6,-3,0,3,6... for the integer set Z_3 so the congruent mod 3.

 

[HallsofIvy]

Then, strictly speaking, each member of Z<sub>3[/sup] is an equivalence class, each class containing exactly one non-negative integer less than 3. That is, there is an equivalence class containing 0, and equivalence class containing 1, and an equivalence class containing 2. That is, I presume, what you mean by \bar{0}, \bar{1}, and \bar{2}. Since that does NOT have 9 members. Am I to assume that you mean the set of all numbers of the form a\alpha+ b where a and b are in Z3 and \alpha satisfies the equation above? That has nine members: 0, 1, 2, \alpha, \alpha+ 1, \alpha+ 2, 2\alpha, 2\alpha+ 1, and 2\alpha+ 2.

Set up your table so it has those both along the top and verticaclly on the left. Multiply each of the 81 pairs and reduce to one of those 9 forms by using the equation \alpha satisfies. For example, \left(\alpha+ 1\right)\left(2\alpha+ 1\right)= 2\alpha^2+ 3\alpha+ 1. Since \alpha^2+ 1= 0, \alpha^2= -1. Of course, 3 is equivalent to 0 mod 3 so this reduces to -2+ 1= -1 which is 2 mod 3:\left(\alpha+ 1\right)\left(2\alpha+ 1\right)= 2.

No, the LaTex editor here does not "suck" but it is a bad idea to try to combine both LaTex and non-LaTex in the same formula: use LaTex for the entire formula.</sub>

 

[phyguy321]

awesome thanks