Hey!!

We have the set $A=\{a_1, a_2, \ldots \}$, the $a_i$'s might be finitely or infinitely many.

We have that $\mathbb{Q}(A)=\left \{\frac{f(a_1, \ldots , a_n)}{g(a_1, \ldots , a_n)} : f,g\in \mathbb{Q}[x_1, \ldots , x_n], g\neq 0, a_1, \ldots , a_n\in A, n\in \mathbb{N}\right \}$.

We have that $\mathbb{Q}(A)=\bigcup_{n=1}^{\infty}P_n$ where $P_n$ is the set of all polynomials of $\mathbb{Q}(A)$ of degree n. $P_n$ can be represented as a $(n+1)$-tuple of the rational coefficients, right?

It holds that $|P_n|=|\mathbb{Q}^{n+1}|=|\mathbb{Q}|$.

Then we have the following:

$$|\mathbb{Q}(A)|=|\bigcup_{n=1}^{\infty}P_n|\leq \sum_{n=1}^{\infty}|P_n|=\sum_{n=1}^{\infty}|\mathbb{Q}|$$

Is the last sum correct?