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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 $...