## A Constructive Proof of the Fundamental Theorem of Algebra without using the Rationals (2001)

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Venue: | In Callaghan et al |

Citations: | 27 - 3 self |

### BibTeX

@INPROCEEDINGS{Geuvers01aconstructive,

author = {Herman Geuvers and Freek Wiedijk and Jan Zwanenburg},

title = {A Constructive Proof of the Fundamental Theorem of Algebra without using the Rationals},

booktitle = {In Callaghan et al},

year = {2001},

pages = {96--111},

publisher = {Springer}

}

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### Abstract

In the FTA project in Nijmegen we have formalized a constructive proof of the Fundamental Theorem of Algebra. In the formalization, we have first defined the (constructive) algebraic hierarchy of groups, rings, fields, etcetera. For the reals we have then defined the notion of real number structure, which is basically a Cauchy complete Archimedean ordered field. This boils down to axiomatizing the constructive reals. The proof of FTA is then given from these axioms (so independent of a specific construction of the reals), where the complex numbers are defined as pairs of real numbers. The proof of FTA that we have chosen to formalize is the one in the seminal book by Troelstra and van Dalen [17], originally due to Manfred Kneser [12]. The proof by Troelstra and van Dalen makes heavy use of the rational numbers (as suitable approximations of reals), which is quite common in constructive analysis, because equality on the rationals is decidable and equality on the reals isn't. In our case, this is not so convenient, because the axiomatization of the reals doesn't `contain' the rationals. Moreover, we found it rather unnatural to let a proof about the reals be mainly dealing with rationals. Therefore, our version of the FTA proof doesn't refer to the rational numbers. The proof described here is a faithful presentation of a fully formalized proof in the Coq system.

### Citations

14 | The algebraic hierarchy of the FTA Project
- Geuvers, Pollack, et al.
- 2002
(Show Context)
Citation Context ...al numbers axiomatically. More precisely, the reals form a part of a constructive algebraic hierarchy, which consists (among other things) of the abstract notions of rings,selds and orderedselds. See =-=[8]-=- for details. The base level of this hierarchy consists of the notion of constructive setoid, which is basically a pair of a type and an apartness relation over the type. (For constructive reals, `bei... |

12 |
The fundamental theorem of algebra
- Fine, Rosenberger
- 1997
(Show Context)
Citation Context ...hat For every non-constant polynomial f(z) = an z n + an 1 z n 1 + : : : + a 1 z + a 0 with coecients in C , the equation f(z) = 0 has a solution. This theorem has a long and illustrious history (see =-=[6]-=- or [11] for an overview). It was proved for thesrst time in Gauss's Ph.D. thesis from 1799. Many proofs of the Fundamental Theorem of Algebra are known, most of which have a constructive version. The... |

10 |
Uniformly convergent algorithms for the simultaneous approximation of all zeros of a polynomial
- Gargantini, Henrici
- 1969
(Show Context)
Citation Context ...l zeros of a (monic) polynomial. A similar but more abstract proof, also using the winding number, occurs in [1], where FTA is proved for arbitrary non-constant polynomials. Based on Weyl's approach, =-=[10]-=- presents an implementation of an algorithm for the simultaneous determination of the zeros of a polynomial. In [2], Brouwer and De Loor give a constructive proof of FTA for monic polynomials bysrst p... |

7 |
Constructive Analysis. Number 279 in Grundlehren der mathematischen Wissenschaften
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...ic polynomials) is from Weyl [18], where the winding number is used to simultaneouslysnd all zeros of a (monic) polynomial. A similar but more abstract proof, also using the winding number, occurs in =-=[1]-=-, where FTA is proved for arbitrary non-constant polynomials. Based on Weyl's approach, [10] presents an implementation of an algorithm for the simultaneous determination of the zeros of a polynomial.... |

7 |
eds.), Constructive aspects of the fundamental theorem of algebra
- Dejon, Henrici
- 1969
(Show Context)
Citation Context ... over the complex numbers, there exists z 2 C such that f(z) = 0. Proof. Take any c > 0 with c > jf(0)j. We construct a sequence z i 2 C such that for all i jf(z i )jsi c (3) jz i+1 z i js(q i c) 1=n =-=(4-=-) where qs1 is given by the Kneser Lemma 4. This sequence is constructed by iteratively applying the Kneser Lemma to f z i (z) f(z+z i ) tosnd z i+1 z i . The required properties of z i then follow d... |

7 |
Erganzung zu einer Arbeit von Hellmuth Kneser uber den Fundamentalsatz der Algebra
- Kneser
- 1981
(Show Context)
Citation Context ...complex numbers are dened as pairs of real numbers. The proof of FTA that we have chosen to formalize is the one in the seminal book by Troelstra and van Dalen [17], originally due to Manfred Kneser [=-=12]-=-. The proof by Troelstra and van Dalen makes heavy use of the rational numbers (as suitable approximations of reals), which is quite common in constructive analysis, because equality on the rationals ... |

4 |
Henrici (editor), Constructive Aspects of the Fundamental Theorem of Algebra
- Dejon, Henrici
- 1967
(Show Context)
Citation Context ...ver the complex numbers, there exists z ∈ C such that f(z) = 0. Proof. Take any c > 0 with c > |f(0)|. We construct a sequence zi ∈ C such that for all i |f(zi)| < q i c (3) |zi+1 − zi| < (q i c) 1/n =-=(4)-=- where q < 1 is given by the Kneser Lemma 4. This sequence is constructed by iteratively applying the Kneser Lemma to fzi (z) ≡ f(z+zi) to find zi+1−zi. The required properties of zi then follow direc... |

2 |
Intuitionistische Ergänzung des Fundamentalsatzes der Algebra
- Brouwer
- 1924
(Show Context)
Citation Context ...plex number as the limit of a series of rational complex numbers) to general monic polynomials over C . This proof { and also Weyl's and other FTA proofs { are discussed and compared in [14]. Brouwer =-=[3]-=- was thesrst to generalize the constructive FTA proof to arbitrary non-constant polynomials (where we just know some coecient to be apart from 0). In [16] it is shown that, for general non-constant po... |

2 |
FTA project, http://www.cs.kun.nl/gi/projects/fta
- Geuvers, Wiedijk, et al.
- 2000
(Show Context)
Citation Context ... using some polynomial arithmetic. Therefore there's no use of linear algebra in the proof anymore. We have formalized the proof presented here using the Coq system: this was known as the FTA project =-=[7]-=-. In the formalization, we treat the real numbers axiomatically. More precisely, the reals form a part of a constructive algebraic hierarchy, which consists (among other things) of the abstract notion... |

2 |
Every polynomial has a root
- Littlewood
- 1941
(Show Context)
Citation Context ...to formalize. (One wouldsrst have to dene the arctan function and establish an isomorphism between the two representations.) Therefore we have chosen a dierent proof, which appears e.g. in [5], and [1=-=3]-=- and is basically constructive. Here, the existence of k-th roots in C is derived directly from the existence of square roots in C and the fact that all polynomials over R of odd degree have a root. T... |

1 |
Loor, Intuitionistischer Beweis des Fundamentalsatzes der Algebra
- Brouwer, de
- 1924
(Show Context)
Citation Context ...ere FTA is proved for arbitrary non-constant polynomials. Based on Weyl's approach, [10] presents an implementation of an algorithm for the simultaneous determination of the zeros of a polynomial. In =-=[2]-=-, Brouwer and De Loor give a constructive proof of FTA for monic polynomials bysrst proving it for polynomials with rational complex coecients (which have the advantage that equality is decidable) and... |

1 |
Fundamentalsatz der Algebra und der
- Kneser, Der
- 1940
(Show Context)
Citation Context ... every non-constant polynomial f(z) = an z n + an 1 z n 1 + : : : + a 1 z + a 0 with coecients in C , the equation f(z) = 0 has a solution. This theorem has a long and illustrious history (see [6] or =-=[11]-=- for an overview). It was proved for thesrst time in Gauss's Ph.D. thesis from 1799. Many proofs of the Fundamental Theorem of Algebra are known, most of which have a constructive version. The proof t... |

1 |
Die Hoofstelling van die Algebra van Intutionistiese standpunt
- Loor
- 1925
(Show Context)
Citation Context ...(viewing a complex number as the limit of a series of rational complex numbers) to general monic polynomials over C . This proof { and also Weyl's and other FTA proofs { are discussed and compared in =-=[14]-=-. Brouwer [3] was thesrst to generalize the constructive FTA proof to arbitrary non-constant polynomials (where we just know some coecient to be apart from 0). In [16] it is shown that, for general no... |

1 |
Ein konstruktiver Beweis des Fundamentalsatzes
- Schwichtenberg
(Show Context)
Citation Context ...rsion of this proof is given, using rational approximations to overcome the undecidability of equality on the reals. Another constructive version of the Kneser proof is presented by Schwichtenberg in =-=[15-=-], also using rational approximations, but along dierent lines. The constructive version of FTA reads as follows. For every polynomial f(z) = an z n + an 1 z n 1 + : : : + a 1 z + a 0 with coecients i... |