Originally Posted by

**james123** I've tried all sorts of things, such as (call my map on D[x]\0 V)

$\displaystyle V(a_0 + a_1x + a_2x^2 + \ldots + a_nx^n) = 2^{n}(v(a_0) + v(a_1) + \ldots + v(a_n))$

(the 2^n is to distinguish a_0 being a unit from a_1, say, being a unit for property (ii)), but I'm always hitting a snag for property (i) since there isn't linearity of v. I know the integral domain $\displaystyle \mathbb{Z}[\sqrt{-5}]$ is an example of such a D, with $\displaystyle v(a+b\sqrt{-5}) = a^2 + 5b^2$ so I've been playing with $\displaystyle (\mathbb{Z}[\sqrt{-5}])[x]$ to help come up with a desirable V, but to no avail.

If you could suggest what the map should like I'd be most appreciative.

Many thanks