Roots of polynomials over semirings and hyperfields

Abstract

We continue our investigation of roots of polynomials over semirings and hyperfields, via ``pairs'' with a surpassing relation $\preceq,$ employing a which we call $\preceq$-reversibility. The ensuing results include a fundamental theorem of algebra for pairs, that tangible polynomials with enough roots ``$\preceq$-split,'' at times uniquely, into linear factors over a suitable finite extension of pairs. We also see that polynomials that agree on ``almost'' all null roots are ``almost'' equal. Finally, we obtain roots of integral polynomials over extension pairs, providing a construction of integrally closed pairs over hyperfields and over zero sum free semirings.