Prove: set of the polynomials of degree $n$ with coefficients in $\Bbb Q $ is countable Let $P_n =\{p(x)=a_n x^n+ a_{n-1} x^{n-1}+...+ a_1 x+a_0 |a_i \in \Bbb Q \}$ the set of the polynomials of degree $n$ with coefficients in $\Bbb Q $
Prove that $P_n$ is countable and tell why $P= \bigcup_{n=0}^\infty P_n$ the set of all the polynomials with coefficient in $\Bbb Q $ is countable.
How  can I prove that $|P_n|=|Q|^{n+1}$ ?
 A: Hint: Note that $\Bbb Q$ is countable and each coefficient (of which there are $n+1$) is from $\Bbb Q$. Now use the fact that a countable union of countable sets is countable.
It might help to note that each polynomial with coefficients in $\Bbb Q$ is equivalent to a polynomial with coefficients in $\Bbb Z$ (why?). If you can find an injection from $\Bbb Z\times\Bbb Z\to\Bbb Z$, you can try to extend the idea to $\Bbb Z\times\cdots\times\Bbb Z$, $(n+1)$-copies of $\Bbb Z$.
A: For $p_n(x)$, a n degree polynomial with coefficients in $\mathbb{Q}$, there are only n+1 coefficients ($a_i$'s). If these n+1 coefficients are fixed, the polynomial is fixed.
Now, total number of such $p_n (x)$ i.e. $|P_n|$ is $|\mathbb{Q}|^{n+1}$   (each coefficient could take any value in $\mathbb{Q}$), which is countable.
Hence, $|P_n|$ is countable.
We know that countable union of countable sets is countable $\implies |P|$ is countable 
A: Hint: you may use the Cantor-Bernstein theorem to prove that:
$$\left| P_n \right| = \left|\mathbb{Q}^n\right| = \left|\mathbb{N}^n\right| = \left|\mathbb{N}\right| = \aleph_0. $$
