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Algebraically closed abelian \(l\)-groups. (English) Zbl 1363.06029

Summary: Every semifield of non-zero characteristic is either a field of prime characteristic or a semifield of characteristic 1. Semifields of characteristic 1 are equivalent to abelian lattice-ordered groups. It is proved that such a semifield \(A\) is algebraically closed if and only if the pure equations \(x^n = a\) and certain quadratic equations are solvable in \(A\). Using a sheaf representation for \(z\)-projectable abelian \(\ell \)-groups on the co-Zariski space of minimal primes, a sheaf-theoretic characterization of algebraic closedness in characteristic 1 is obtained. Concerning the solvability of quadratic equations, the criterion consists in a topological condition for the base space. The results are built upon an analysis of rational functions in characteristic 1. While polynomials satisfy the “fundamental theorem of algebra”, the multiplicative structure of rational functions is determined by means of “divisors”, extracted from the additive structure of \(A\) modulo parallelogram identities.

MSC:

06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
12K10 Semifields
16Y60 Semirings
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