Let R be a domain Prove that if a polynomial in Rx is a unit

Let R be a domain. Prove that if a polynomial in R[x] is a unit, then it is a nonzero constant (the converse is true if R is a field). Show that (2x + 1)^2 = 1 in Z_4[x]. Conclude that 2x + 1 is a unit in Z_4[x], and that the hypothesis in part (i) that R be a domain is necessary.

Solution

(i) Let R be a domain.

If f(x) is a unit in R[x] , there exists a g(x) in R[x] with

                                f(x)g(x)=1............................(1)

        Let f(x) = a[k]x^k+.......and f be of degreee k. So a[k] not zero

             g(x) = b[m]x^m +..............and g be of degree m. So b[m] not zero.

As R is a domain , a[k]b[m] is not zero in R and hence degree (fg) is not zero.,

So (1) is possible only if f(x) is of degree 0 and is a non-zero constant.

If R is a field , any non-zero element in R is a unit and the claim follows.

(II) (2X+1)²= 4X²+4X+1 = 1 and hence (2X+1) (a non-constant polynomial) is a unit in Z₄[X].

(This is explained by the fact that Z₄ is not a domain.