Let $V$ be an irreducible affine variety over a finite field $\mathbb{F}_q$, , given in terms of equations over $\mathbb{F}_q$, where $q$ is some prime power. Are there any methods to decide whether $V$ remains irreducible over the algebraic closure $\overline{\mathbb{F}_q}$?
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$\begingroup$ How did you check it was irreducible over $\mathbb F_q$? $\endgroup$– Will SawinJul 31, 2021 at 15:11
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$\begingroup$ For two variables case, I used Eisenstein s criterion $\endgroup$– VanyaJul 31, 2021 at 15:15
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3$\begingroup$ Why not apply Eisenstein's criterion over $\overline{\mathbb F_q}$? $\endgroup$– Will SawinJul 31, 2021 at 15:18
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$\begingroup$ Affine variety is defined by several equations. Eisenstein's criterion is very limited in application. $\endgroup$– VanyaJul 31, 2021 at 15:59
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1$\begingroup$ But how is the base field relevant for this? If it works over $\mathbb F_q$ then it works over $\overline{\mathbb F_q}$. $\endgroup$– Will SawinJul 31, 2021 at 16:23
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