Let \[ f(x)=x^5-8x^3+9x-3, \qquad g(x)=x^4-5x^2-6x+3. \] Prove that there is an integer \(d\) such that \(f\) and \(g\) have a common root in \[ \mathbb Q(\sqrt d). \] Determine \(d\).
Solution
1. The two polynomials have the common quadratic factor\[ q(x)=x^2-3x+1. \]
Direct multiplication gives \[ \begin{aligned} (x^2-3x+1)(x^3+3x^2-3) &= x^5-8x^3+9x-3\\ &= f(x), \end{aligned} \] and \[ \begin{aligned} (x^2-3x+1)(x^2+3x+3) &= x^4-5x^2-6x+3\\ &= g(x). \end{aligned} \] Hence every root of \(q\) is a common root of \(f\) and \(g\).
2. The roots of \(q\) are\[ \frac{3+\sqrt5}{2} \qquad\text{and}\qquad \frac{3-\sqrt5}{2}. \]
The quadratic formula applied to \[ x^2-3x+1=0 \] gives \[ x = \frac{3\pm\sqrt{9-4}}2 = \frac{3\pm\sqrt5}{2}. \]
3. The required integer is\[ \boxed{d=5}. \]
By steps 1 and 2, both \(f\) and \(g\) vanish at \[ \frac{3+\sqrt5}{2}\in\QQ(\sqrt5). \] Thus \(d=5\) has the required property. The other root lies in the same quadratic field.
4. Q.E.D.
Step 3 gives the required quadratic field and integer.