Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 3 / 10E (Groups, Rings and Modules)

The content $c(f)$ is a greatest common divisor of the coefficients, defined up to a unit. A polynomial is primitive when its content is a unit.

If a prime divided every coefficient of $fg$, reducing modulo that prime would make the product of the two nonzero reduced primitive polynomials zero in a domain, impossible. Thus a product of primitive polynomials is primitive. Factoring the contents from arbitrary $f,g$ then givesfor a unit $u$.

Write the rational root in lowest terms as $a/b$. The factor $bx-a$ is primitive and divides $f$ over $\mathbb{Q}[x]$; Gauss's lemma makes it divide in $\mathbb{Z}[x]$. Comparison of leading coefficients makes $b$ divide the leading coefficient one, so $b=\pm 1$ and the root is integral.

Regard the polynomial as an element of $\mathbb{C}[x][y]$:The prime $x-1$ divides every nonleading coefficient, its square does not divide the constant coefficient, and it does not divide the leading coefficient $x^3$. Eisenstein's criterion proves irreducibility over $\mathbb{C}(x)[y]$; the polynomial is primitive in $\mathbb{C}[x][y]$, so Gauss's lemma proves irreducibility there.