irreducible polynomials obtained from biquadratic fields

Corollary.

Let $m$ and $n$ be distinct squarefree integers, neither of which is equal to $\mathrm{1}$. Then the polynomial

$$x^4-2(m+n)x^2+{(m-n)}^2$$

is irreducible (http://planetmath.org/IrreduciblePolynomial2) (over $\mathrm{Q}$).

Proof.

By the theorem stated in the parent entry (http://planetmath.org/PrimitiveElementOfBiquadraticField), $\sqrt{m}+\sqrt{n}$ is an algebraic number of degree (http://planetmath.org/DegreeOfAnAlgebraicNumber) $4$. Thus, a polynomial of degree $4$ that has $\sqrt{m}+\sqrt{n}$ as a root must be over $\mathbb{Q}$. We set out to construct such a polynomial.