A question regarding properties of Algebraic Integers

Question

I am trying questions of Field theory from Thomas Hungerford and couldn't solve this question exercise 18 of Section 5.1 .

I have done (a), (b) but couldn't do any of parts of (c) , (d) .

I tried using definition of algebraic integer by assuming a polynomial p(x) in $\mathbb{Z}(x) $ and trying to prove that there will exist an polynomial in $\mathbb{Z}(x) $ with required properties but couldn't for any parts of (c) and (d).

Can you please tell how to solve any part of (c) or(d) . Rest I would like to try by myself.

Thanks!!

Answer 1

For the part (c), consider $P(X-n)$ and $n^d P(\frac{X}{n})$ where $P$ is a monic polynomial of $\mathbb{Z}[X]$ vanishing $u$ and $d$ its degree.

The results of the question (d) is well-known but I don't know if there is simple answer. The classical proof consists in proving that $u$ is integral over $\mathbb{Z}$ if and only if the subring $\mathbb{Z}[u]$ is finitely generated over $\mathbb{Z}$. There exists an alternative proof based on the resultant which produce an explicit equation, see the question How to prove that the sum and product of two algebraic numbers is algebraic?.