How to prove $\left| P_n \right| = \left|\mathbb{Q}^n\right|$ Where $P_n =\{p(x)=a_n x^n+ a_{n-1} x^{n-1}+...+ a_1 x+a_0 |a_i \in \Bbb Q \}$.
This is, the set of polynomial of degree $n$ and coefficient in $\mathbb{Q}$
I was thinking in this: For example, How many $4$ digits number are there? In a $4$ digit number, we have $4$ places to fill which can be filled by only $0,1,…,9$. So, number of $4$ digits number are 9*10*10*10 (since first place cannot be zero) Similarly, i have n+1 places (coefficients) and |Q| ways to fill them.
But, formally i cannot find a biyection, or give a formal proof of this. Can someone help me?
Hint: Consider $P_n \to \mathbb Q^n$ given by $p \mapsto (a_{n-1}, \dots a_1, a_0 )$.