Past exam of the mathematics course of the University of Cambridge / 2005 / ib / Paper 3 / 11C / ii / Solution

The polynomial $p=x^5-4x+2$ satisfies the Eisenstein criterion at the prime two: every nonleading coefficient is even, the leading coefficient is not, and the constant term is not divisible by four. Hence it is irreducible over $\mathbb{Q}$. Its ideal is maximal in $\mathbb{Q}[x]$; explicitly, for any polynomial not divisible by $p$, the Euclidean algorithm gives $ag+bp=1$, furnishing the inverse of its residue class. ThusFor the integer quotient, division by the monic polynomial $p$ gives a unique remainder of degree below five with integer coefficients. If an integer polynomial becomes zero in the rational quotient, its integer remainder is still divisible by $p$ over $\mathbb{Q}$, so must be zero. This gives an injective ring map $\mathbb{Z}[x]/(p)\to \mathbb{Q}[x]/(p)$, proving the monic irreducible integer-polynomial quotient is a domain.

The same unique-remainder argument makes the integer quotient a free abelian group with basis $1,x,x^2,x^3,x^4$. The class of two is nonzero and cannot have an inverse: twice an integer remainder cannot equal the remainder one. Consequently