1 - Field Extensions and Differential Field Extensions

Abstract

(I) The problem of “solving algebraic equations by radicals” (where by algebraic equations we mean those of the form f (x) = 0, for f a non-zero polynomial with rational coefficients) was one of the most important topics in mathematics from the earliest periods of antiquity until the 19th century. The Babylonians, and later the Greeks, already knew how to solve quadratic equations, although their formulas were more complex than those we use today, since their notation was inferior and they lacked the concepts of zero and negative numbers. Cubic equations were solved by S. del Ferro (around 1515) and N. Tartaglia (whose contributions were published in 1545 by G. Cardano and are often incorrectly attributed to the latter); quartic equations were solved by L. Ferrari using a method that was also published in Cardano’s Ars Magna together with the method proposed by Tartaglia. In 1576, R. Bombelli compiled a summary of all of this work using simpler notation in Algebra. However, all efforts to solve quintic equations were unsuccessful from the end of the 16th century until the early 19th century. J.-L. Lagrange was arguably the first to recognize the underlying reasons in his memoirs from 1770–1771. Finally, P. Ruffini attempted to show that the general quintic equation cannot be solved by radicals in a series of dense and controversial memoirs gradually published between 1799 and 1813. One proof that was entirely correct but weighed down by long calculations was provided by N. Abel in 1824; in 1826, he showed.