Polynomials with integer coefficients - the GCD identity

I am reading Anderson and Feil on the Factorization of Polynomials - Section 5.3 Polynomials with Integer Coefficients

A&F point out that some of the therems they have proved for are false if we restrict ourselves to polynomials with integer coefficients.

For example they point out that the Divsion Theorem is false for .

Also the GCD identity fails in .

They then ask the reader the polynomials 2 and x in pointing out the 1 is the GCD for these polynomials. A&F then ask the reader to prove that we cannot write 1 as a linear combination of 2 and x

Can anyone help me with a rigorous and formal proof of this.

Peter

Re: Polynomials with integer coefficients - the GCD identity

suppose we COULD.

we then have 1 = 2f(x) + xg(x), for some polynomials f(x),g(x) in Z[x].

suppose that deg(f) = m, and deg(g) = n. let k = max(m,n). then we can write:

(some of the coefficients may be 0).

then:

.

since these are equal polynomials, we must have equal constant terms, so for some integer , which cannot happen.

Re: Polynomials with integer coefficients - the GCD identity

Excellent ... yes, so obvious when you see how ... :-)

Thanks so much ....

Peter