Minimal fields (Q2785963)

The notions of a minimal subset of a structure and of a strongly minimal subset were introduced and studied in [\textit{W. E. Marsh}, On \(\omega_1\)-categorical and not \(\omega\)-categorical theories, Dissertation, Dartmouth College (1966) and \textit{J. T. Baldwin} and \textit{A. H. Lachlan}, On strongly minimal sets, J. Symb. Log. 36, 79-96 (1971; Zbl 0217.30402)]. It is possible to define the dimension of a minimal structure similar to the definition of the dimension of a vector space. The problem is how to characterize minimal groups, rings, and fields in algebraic terms. It is known that a field is strongly minimal iff it is algebraically closed. But it is an open question whether a minimal field is algebraically closed. NEWLINENEWLINENEWLINEMinimal fields are studied in the paper under review. The multiplicative group of a minimal field is divisible. Every polynomial of degree less than 5 is reducible over a minimal field, as well as polynomials of the form \(x^n-a.\) It is proved that a Henselian algebraic extension of a minimal field is algebraically closed. One of the consequences of the fact is a necessary condition for the existence of irreducible polynomials of \(n\)-th degree. Every polynomial is reducible over a minimal field if its degree is not equal to 2 or 5.