I have long interpreted Gauss’s lemma to mean that if a polynomial with integral coefficients has a rational root, that root has to be an integer.
This is incorrect.
For example, take the polynomial . It has integral coefficients. However, it does not have an integral root. It has a rational root; namely .
Gauss’s lemma just states that a polynomial with integral coefficients, if it can be factored into polynomials with rational coefficients, can be factored into polynomials with integral coefficients.
The rational roots theorem simply follows from this.
The proof has been omitted. But it is quite simple.