What is the degree of the simplest polynomial with the zeroes:√2, 2√2 + i, and 2 + 2i ??
This means that $\displaystyle x - \sqrt{2}$, $\displaystyle x - (2 \sqrt{2} + \pi)$, $\displaystyle x - (2 + 2 \pi)$ are all factors of this polynomial. Therefore the simplest polynomial that has all these roots is :
$\displaystyle (x - \sqrt{2})(x - (2 \sqrt{2} + \pi))(x - (2 + 2 \pi)) = 0$
Or, equivalently :
$\displaystyle (x - \sqrt{2})(x - 2 \sqrt{2} - \pi)(x - 2 - 2 \pi) = 0$
And the degree of this polynomial is, trivially, 3 (can be checked by expanding).