# Note on generalizing theorems in algebraically closed fields

Baaz, Matthias, and Richard Zach. 1998. “Note on Generalizing Theorems in Algebraically Closed Fields.” Archive for Mathematical Logic 37 (5–6): 297–307. https://doi.org/10.1007/s001530050100.

The generalization properties of algebraically closed fields $$\mathit{ACF}_p$$ of characteristic $$p > 0$$ and $$\mathit{ACF}_0$$ of characteristic 0 are investigated in the sequent calculus with blocks of quantifiers. It is shown that $$\mathit{ACF}_p$$ admits finite term bases, and $$\mathit{ACF}_0$$ admits term bases with primality constraints. From these results the analogs of Kreisel’s Conjecture for these theories follow: If for some $$k$$, $$A(1 + … + 1)$$ ($$n$$ 1’s) is provable in $$k$$ steps, then $$(\forall x)A(x)$$ is provable.

Review: Yehuda Rav (Mathematical Reviews 2000a:03057)