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We use the definitions of this question. Let $V$ be a Zariski closed subset of $\Omega^n$. Let $V_1,\dots,V_r$ be all the irreducible components of $V$. Let $k$ be a common field of definition of all $V_i$. Is $k$ a field of definition of $V$?

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If $V$ and $W$ are closed subsets of $\mathbf{A}^n_{K}$ and $k$ is a common field of definition for $V$ and $W$, it is easy to see that $k$ is a field of definition for $V\cup W$. You can check this locally. Then, it boils down to the statement that if $I$ and $J$ are ideals in $K[x_1,\ldots,x_n]$ which can be generated by elements in $k[x_1,\ldots,x_n]$, then $I\cdot J$ is also generated by such elements.