Let k be a domain and let fx k x If ax is an associate of f

Let k be a domain and let f(x) k [x] If a(x) is an associate of f, prove deg(f) = deg(a). Give an example to show that the statement may be false if is not a domain.

Solution

If a(x) is an associate of f(x) , then

a(x)= u f(x), where u is a unit in k[x].

But the only units in k[x] are the units of k.

So the degree of a(x) = degree of f(x).

Consider Z₄[x]. Let f(x) = x. (degree f =1). Now u= 1+2x is a unit in Z₄[x](it is its own inverse).

a(x) =f(x)u is clearly an associate of f(x) and is of degree 2.

This shows that the statement may be false if k is not a domain