# Real-algebraic/Q or Z/Exercise

Let be a real number. Show that the following properties are equivalent.

- There exist a polynomial , , with integer coefficients and with .
- There exists a polynomial , , wit .
- There exists a normed polynomial with .

Let ${}z\in \mathbb {R}$ be a real number. Show that the following properties are equivalent.

- There exist a polynomial ${}P\in \mathbb {R} [X]$, ${}P\neq 0$, with integer coefficients and with ${}P(z)=0$.
- There exists a polynomial ${}Q\in \mathbb {Q} [X]$, ${}Q\neq 0$, wit ${}Q(z)=0$.
- There exists a normed polynomial ${}R\in \mathbb {Q} [X]$ with ${}R(z)=0$.