# Binary Operation

1. Nov 13, 2005

### Icebreaker

How is multiplication in $$R=\mathbb{Z}_5 \times \mathbb{Z}_5$$ defined? if (a,b) and (c,d) is in R, what's (a,b)(c,d)? (ac,bd)?

2. Nov 13, 2005

### Muzza

Usually pointwise (i.e. (a, b)(c, d) = (ac, bd) as you guessed). It can easily be extended to other groups (rings).

3. Nov 13, 2005

### Icebreaker

What do you mean by extending to other rings? I'm trying to find an isomorphism between Z5XZ5 and Z5[x]/X^2+1 and am having a hard time finding it. If I can redefine multiplication in Z5XZ5 then it will be easy.

4. Nov 13, 2005

### Muzza

I mean that given any rings G, H, you can easily define the product ring GxH in the same (pointwise) fashion.

I doubt the author (unless he or she said otherwise) intended for you to redefine Z_5 x Z_5.

Given any polynomial p, there are unique constants a, b and a polynomial q such that

p(x) = q(x)(x^2 + 1) + ax + b.

There seems to be an obvious function between Z_5[x]/(x^2 + 1) and Z_5 x Z_5 to try. But I haven't myself.