## The fundamental theorem of algebra: a constructive development without choice (2000)

Venue: | Pacific Journal of Mathematics |

Citations: | 12 - 3 self |

### BibTeX

@ARTICLE{Richman00thefundamental,

author = {Fred Richman},

title = {The fundamental theorem of algebra: a constructive development without choice},

journal = {Pacific Journal of Mathematics},

year = {2000},

volume = {196},

pages = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

Is it reasonable to do constructive mathematics without the axiom of countable choice? Serious schools of constructive mathematics all assume it one way or another, but the arguments for it are not compelling. The fundamental theorem of algebra will serve as an example of where countable choice comes into play andhow to proceedin its absence. Along the way, a notion of a complete metric space, suitable for a choiceless environment, is developed. By constructive mathematics I mean, essentially, mathematics that is developed along the lines proposed by Errett Bishop [1]. More precisely, I mean mathematics that is done in the context of intuitionistic logic — without the lawof excluded middle. My reasons for identifying these notions are discussed in [9] and [10], the basic contention being that constructive mathematics has the same subject matter as classical mathematics. Ruitenburg [11] treated the fundamental theorem of algebra in a choiceless

### Citations

417 | Foundations of Constructive Analysis
- Bishop
- 1967
(Show Context)
Citation Context ... a complete metric space, suitable for a choiceless environment, is developed. By constructive mathematics I mean, essentially, mathematics that is developed along the lines proposed by Errett Bishop =-=[1]-=-. More precisely, I mean mathematics that is done in the context of intuitionistic logic — without the lawof excluded middle. My reasons for identifying these notions are discussed in [9] and [10], th... |

25 |
Zermelo’s Axiom of Choice. Its Origins, Development and Influence
- Moore
- 1982
(Show Context)
Citation Context ...tegers to sequences that represent the same real number. Presumably that problem does not arise with natural numbers because they have canonical presentations. But Lebesgue said, in a letter to Borel =-=[8]-=-, “I agree completely with Hadamard when he states that to speak of an infinity of choices without giving a rule presents a difficulty that is just as great whether or not the infinity is denumerable.... |

17 | Lectures on constructive mathematical analysis, volume 60 of Translations of Mathematical Monographs - Kushner - 1984 |

8 | Intuitionism as Generalization
- Richman
- 1990
(Show Context)
Citation Context ...rrett Bishop [1]. More precisely, I mean mathematics that is done in the context of intuitionistic logic — without the lawof excluded middle. My reasons for identifying these notions are discussed in =-=[9]-=- and [10], the basic contention being that constructive mathematics has the same subject matter as classical mathematics. Ruitenburg [11] treated the fundamental theorem of algebra in a choiceless env... |

4 |
Constructing roots of polynomials over the complex numbers. Computational aspects of Lie group representations and related topics
- Ruitenburg
- 1990
(Show Context)
Citation Context ...ddle. My reasons for identifying these notions are discussed in [9] and [10], the basic contention being that constructive mathematics has the same subject matter as classical mathematics. Ruitenburg =-=[11]-=- treated the fundamental theorem of algebra in a choiceless environment. He proved the theorem for real numbers that are defined by Cauchy sequences of rational numbers. An awkward feature of restrict... |

2 |
Intuitionistische Ergänzung des Fundamentalsatzes der Algebra
- Brouwer
- 1924
(Show Context)
Citation Context ...a single root. 7. Homogenizing. What about the restriction to monic polynomials? Bishop, after all, proved that any nonconstant polynomial with complex coefficients has a complex root, as did Brouwer =-=[2]-=-. In the general case, where the formal leadings226 FRED RICHMAN coefficient might be zero, we really need to pass to the complex projective line (the Riemann sphere), and homogeneous polynomials in t... |

2 |
Separably real closed local rings
- Joyal, Reyes
- 1986
(Show Context)
Citation Context ...n a neighborhood of zero. The same phenomenon occurs with respect to the intermediate value theorem for real polynomials, which implies that any cubic polynomial has as218 FRED RICHMAN real root (see =-=[5]-=-). Fix a small positive number a and consider the polynomial pb(x) =2x 3 +3ax 2 + b which has a local maximum at −a and a local minimum at 0. Try to construct a continuous function r on (−a, a) so tha... |

2 |
Sets as limits
- Stolzenberg
- 1988
(Show Context)
Citation Context ...Sn, and an element x in X, then the set of q ∈ Q such that q<d(x, y) for all y ∈ Sm for sufficiently large m, defines a real number f(x) as Dedekind cut, and the function f is a location. Stolzenberg =-=[12]-=- suggested a similar way to specify an element ξ of the completion of a metric space X. Construct a set Σ consisting of pairs (x, c) where x is in X and c is a nonnegative real number — think of x as ... |

2 | Intuitionistic Extensions of the Reals. Nieuw Archief voor Wiskunde - Troelstra - 1980 |

1 |
de Loor, Intuitionistischer Beweis des Fundamentalsatzes der Algebra
- andB
- 1924
(Show Context)
Citation Context ...ruct a root of it in the completion of k by Newton’s method [7, Theorem XII.3.1, p. 295]. By considering winding numbers, you can showthat it must get arbitrarily small, as Brouwer and de Loor did in =-=[3]-=-. Clearly the map from Mn(A) toπn(A) is uniformly continuous on bounded subsets. To prove the uniform continuity in the other direction, we will use a simple geometric lemma, which we give in a very c... |

1 |
der Corput, On the fundamental theorem of algebra
- van
- 1946
(Show Context)
Citation Context ...to the complex projective line (the Riemann sphere), and homogeneous polynomials in two variables. A polynomial whose formal leading coefficient is zero has a root at infinity. Indeed, van der Corput =-=[4]-=-, following Brouwer [2], showed that you could factor an arbitrary nonzero polynomial f(X) =a0 + a1X + ···+ anX n as f(X) =a(X − r1) ···(X − rm)(1 − rm+1X) ···(1 − rnX) where m is any integer between ... |

1 | A course in constructive algebra - Ruitenburg - 1988 |

1 | Troelstra andDirk van Dalen, Constructivism in mathematics: An introduction, North-Holland - S - 1988 |