 Solution

ID: past-exam-of-the-number-theory-course-of-the-university-of-oxford/2007/a/1/iii/solution

The sum of squared roots is

c_{1}^{2} - 2 c_{2}

. The sum of their pairwise products is

(α_{1} α_{2} + α_{2} α_{3} + α_{3} α_{1})^{2} - 2 α_{1} α_{2} α_{3} (α_{1} + α_{2} + α_{3}) = c_{2}^{2} - 2 c_{1} c_{3} .

Their product is

c_{3}^{2}

. Thus the required monic polynomial is

X^{3} + (2 c_{2} - c_{1}^{2}) X^{2} + (c_{2}^{2} - 2 c_{1} c_{3}) X - c_{3}^{2} .

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