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 $|P_n|=|Q|^{n+1}$ ?
Best Answer
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 $\Bbb Z$.