The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. ...

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How is this fact implicated by the hypotheses?

Let $K$ be a field and $\overline{K}$ its algebraic closure, then we define the $n$-dimensional space $$\mathbb{A}^{n}=\{(x_1, \ldots, x_{n})\mid x_1, \ldots, x_{n} \in \overline{K}\}$$ So we ...