The Disquisitiones Arithmeticae and the Theory of Equations

Abstract

Carl Friedrich Gauss enriched the theory of algebraic equations with his four proofs of the Fundamental Theorem of Algebra – see [Netto 1913] –, but also with the Disquisitiones Arithmeticae (abbreviated in what follows by D.A.). Our goal is to discuss sec. 7 of the D.A., “On the equations on which the division of the circle depends.” This section, which comprises 32 articles, is entirely devoted to the equations xn − 1 = 0 (that is, what we call cyclotomy theory) and to some striking number-theoretic implications. It uses the basic concepts of the theory of equations at that time: roots and their relations, resolvents. However, our main thesis is that Gauss’s cyclotomy theory marks a turning point in the theory of equations since it brought to the fore the concept of irreducibility. More precisely, Gauss insisted on the systematic search for “equations of as low an order as possible” satisfied by a given quantity. Such equations cannot be factorized into equations of lower degree with respect to a given domain of rationality; moreover, Euclidean algorithm shows that, for a given quantity, they are uniquely determined up to constant factors. Such equations, for which Gauss apparently did not introduce a specific name, were called irreducible in the writings of Niels Henrik Abel and Evariste Galois, see [Abel 1829], § 1, and [Galois 1831/1846].