Rump, Wolfgang Algebraically closed abelian \(l\)-groups. (English) Zbl 1363.06029 Math. Slovaca 65, No. 4, 841-862 (2015). 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. Cited in 1 Document MSC: 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 12K10 Semifields 16Y60 Semirings Keywords:\(\ell \)-group; semifield; Newton polygon; sheaf; Stone space PDFBibTeX XMLCite \textit{W. Rump}, Math. Slovaca 65, No. 4, 841--862 (2015; Zbl 1363.06029) Full Text: DOI References: This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.