In $\mathbb{N}$, we can define an equivalence relation on the Cartesian product $\mathbb{N}^2$ as $(a,b) \sim (c,d)$ if and only if $a + d = b + c$. Then, the quotient set $\mathbb{N}^2 / \sim$ is isomorphic to $\mathbb{Z}$ (this is one way $\mathbb{Z}$ is defined in some books).

Similarly, there is an equivalence relation on the set of all Cauchy sequences in $\mathbb{Q}$, denoted $\mathbb{L}$, where $p_n \sim q_n$ if and only if $\lim_{n \to \infty}(p_n - q_n) = 0$. The quotient set $\mathbb{L} / \sim$ is isomorphic to $\mathbb{R}$.

My question is: Is there an equivalence relation on the $n$-fold Cartesian product $\mathbb{Q}^n$ such that the quotient set $\mathbb{Q}^n / \sim$ is isomorphic to the algebraic numbers $\mathbb{A}$, which are the roots of polynomials with integer coefficients? (I believe $n$ can be finite because $\mathbb{A}$ is countable.)

Furthermore, can such an equivalence relation be defined by some binary operation as in the first example? If so, is this binary operation commutative and associative?

Clarification: By “isomorphic to,” I mean isomorphic as a group. My consideration is: Is there a binary operation on $\mathbb{R}$ that makes $\mathbb{A}$ the smallest field closed under this operation, similar to how $\mathbb{Z}$ is closed under subtraction and $\mathbb{R}$ is the closure under limits, so that I can naturally extend $\mathbb{Q}$ into $\mathbb{A}$?