Book contents
- Frontmatter
- Contents
- Preface
- Part one The Kronecker – Duval Philosophy
- 1 Euclid
- 2 Intermezzo: Chinese Remainder Theorems
- 3 Cardano
- 4 Intermezzo: Multiplicity of Roots
- 5 Kronecker I: Kronecker's Philosophy
- 6 Intermezzo: Sylvester
- 7 Galois I: Finite Fields
- 8 Kronecker II: Kronecker's Model
- 9 Steinitz
- 10 Lagrange
- 11 Duval
- 12 Gauss
- 13 Sturm
- 14 Galois II
- Part two Factorization
- Bibliography
- Index
12 - Gauss
from Part one - The Kronecker – Duval Philosophy
Published online by Cambridge University Press: 15 October 2009
- Frontmatter
- Contents
- Preface
- Part one The Kronecker – Duval Philosophy
- 1 Euclid
- 2 Intermezzo: Chinese Remainder Theorems
- 3 Cardano
- 4 Intermezzo: Multiplicity of Roots
- 5 Kronecker I: Kronecker's Philosophy
- 6 Intermezzo: Sylvester
- 7 Galois I: Finite Fields
- 8 Kronecker II: Kronecker's Model
- 9 Steinitz
- 10 Lagrange
- 11 Duval
- 12 Gauss
- 13 Sturm
- 14 Galois II
- Part two Factorization
- Bibliography
- Index
Summary
Aequationes […] solvere oportebit
C.F. Gauss, Disquisitiones arithmeticaeThis 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.
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- Solving Polynomial Equation Systems IThe Kronecker-Duval Philosophy, pp. 232 - 262Publisher: Cambridge University PressPrint publication year: 2003