The problem of solving polynomials is one of the oldest in mathematics. Following Abel’s proof of the impossibility of solving the quintic in radicals, one must ask: how can we best solve equations of higher degree? Hamilton, Kronecker, Klein, and Hilbert brought into view a rich algebraic landscape, still only partially explored, encompassing the modern invariants of essential dimension and resolvent degree and ripe for future discovery. Hilbert and Arnold called us to push beyond this, toward analysis and topology, with the potential to cast the enduring algebraic problems into new and brighter light.
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