1. direct product of rings

I am stuck on this question:

Show that when is any field in which .

I don't know how to prove that a field is isomorphic to direct product.
What properties do I need to use to prove it?

Can anyone kindly give me some instructions how to do this question.?

Thanks alot

2. Originally Posted by knguyen2005
I am stuck on this question:

Show that when is any field in which .

I don't know how to prove that a field is isomorphic to direct product.
What properties do I need to use to prove it?

Can anyone kindly give me some instructions how to do this question.?
The ring $\mathbb{F}(x)/(x^2-1)$ consists of elements of the form a+bx (where $a,b\in\mathbb{F}$), with algebraic operations $(a+bx) + (c+dx) = (a+b) + (c+d)x$ and $(a+bx)(c+dx) = (ac+bd) + (ad+bc)x$. You have to think of a map from $\mathbb{F}(x)/(x^2-1)$ to $\mathbb{F}\times\mathbb{F}$ that preserves these operations.

The obvious map $a+bx\mapsto(a,b)$ preserves addition but not multiplication (because $(a,b)(c,d) = (ab,cd) \ne (ac+bd,ad+bc)$). So you need to think of a different way of mapping $\mathbb{F}(x)/(x^2-1)$ to $\mathbb{F}\times\mathbb{F}$. I can't say more than that without giving away the whole solution.