A compactness theorem for linear equations (Q2563454)

The Boolean prime ideal theorem (BPI) is known to be equivalent to the assertion that a system \(\Sigma\) of equations over a finite field \(F\) has a solution if each finite subsystem has one. For the field of real numbers the assertion is known to be false for countable \(\Sigma\). \textit{A. Abian} [Arch. Math., Brno 12, 43-44 (1976; Zbl 0351.15004)] has shown that for arbitrary fields the assertion holds if \(\Sigma\) consists of linear equations only. Seemingly unaware of this paper, the authors reproduce Abian's result and counterexample.