This chapter is devoted to two of Gauss' important contributions to solving:

• 12.1 is devoted to a proof of the Fundamental Theorem of Algebra: I use the second proof by Gauss, which is the most algebraic of his four proofs;

• 12.2 presents a résumé of the Disquisitiones Arithmeticae's section devoted to the solution of the cyclotomic equation: I consider these results to be the best pages of Computational Algebra, and I hope to be able to transmit my feeling to the reader.

These two sections also play the rôle of introducing the arguments discussed in the last two chapters: the generalization of Kronecker's Method to real algebraic numbers and Galois Theory.

The Fundamental Theorem of Algebra

In order to present a proof of the Fundamental Theorem of Algebra, and, mainly, to give a statement and a proof which can be easily generalized to an interesting setting (real closed fields), I must start by discussing the elementary and well-known difference between ℝ and ℂ, i.e. that one is ‘ordered’ and the other not:

Definition 12.1.1.A field K is said to be ordered if there is a subset P ⊂ K, the positive cone, which satisfies the following conditions:

Obviously the definition generalizes the trivial property of the ‘positiveness’ relation over ℝ where P is the set of the positive numbers,

In this generalization, it is clearer if we work in the other way: a positive cone P ⊂ K induces on K the total ordering < P defined by:

From now on I will write < omitting the dependence on P.