Give it a thought - (2)!

Moderator note:

Using that as the guiding example, can you come up with a simpler counter example?

Hint: If $\alpha^2 - 3 \alpha + 1 = 0$ , then we have $1 0 1 _ \alpha = 30 \alpha$ . However, a (valid) consideration here is that $3 > \alpha$ , and so $3 0 \alpha$ doesn't necessarily exist (It's debatable, but I tend to prefer that the digits are strictly less than the base).