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?

  • $\begingroup$ There is a bijection from $ \Bbb N$ to $\Bbb Q^n$ or to $\Bbb N^n$ for any positive integer $n.$ Search this site. I suggest searching "cardinal of $\Bbb Q^n\;$" and similar phrases. $\endgroup$ – DanielWainfleet Apr 9 '18 at 21:08

Hint: Consider $P_n \to \mathbb Q^n$ given by $p \mapsto (a_{n-1}, \dots a_1, a_0 )$.


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