The fundamental theorem of algebra states that every non-constant

9 min readMay 29, 2021

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.

Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.

The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division.

Despite its name, there is no purely algebraic proof of the theorem, since any proof must use some form of the analytic completeness of the real numbers, which is not an algebraic concept.[1] Additionally, it is not fundamental for modern algebra; its name was given at a time when algebra was synonymous with theory of equations.

Contents
1 History
2 Proofs
2.1 Complex-analytic proofs
2.2 Topological proofs
2.3 Algebraic proofs
2.3.1 By Induction
2.3.2 From Galois Theory
2.4 Geometric proofs
3 Corollaries
4 Bounds on the zeros of a polynomial
5 See also
6 References
6.1 Citations
6.2 Historic sources
6.3 Recent literature
7 External links
History
Peter Roth, in his book Arithmetica Philosophica (published in 1608, at Nürnberg, by Johann Lantzenberger),[2] wrote that a polynomial equation of degree n (with real coefficients) may have n solutions. Albert Girard, in his book L’invention nouvelle en l’Algèbre (published in 1629), asserted that a polynomial equation of degree n has n solutions, but he did not state that they had to be real numbers. Furthermore, he added that his assertion holds “unless the equation is incomplete”, by which he meant that no coefficient is equal to 0. However, when he explains in detail what he means, it is clear that he actually believes that his assertion is always true; for instance, he shows that the equation {\displaystyle x^{4}=4x-3,}{\displaystyle x^{4}=4x-3,} although incomplete, has four solutions (counting multiplicities): 1 (twice), {\displaystyle -1+i{\sqrt {2}},}{\displaystyle -1+i{\sqrt {2}},} and {\displaystyle -1-i{\sqrt {2}}.}{\displaystyle -1-i{\sqrt {2}}.}

As will be mentioned again below, it follows from the fundamental theorem of algebra that every non-constant polynomial with real coefficients can be written as a product of polynomials with real coefficients whose degrees are either 1 or 2. However, in 1702 Leibniz erroneously said that no polynomial of the type x4 + a4 (with a real and distinct from 0) can be written in such a way. Later, Nikolaus Bernoulli made the same assertion concerning the polynomial x4 − 4x3 + 2x2 + 4x + 4, but he got a letter from Euler in 1742[3] in which it was shown that this polynomial is equal to

{\displaystyle \left(x^{2}-(2+\alpha )x+1+{\sqrt {7}}+\alpha \right)\left(x^{2}-(2-\alpha )x+1+{\sqrt {7}}-\alpha \right),}{\displaystyle \left(x^{2}-(2+\alpha )x+1+{\sqrt {7}}+\alpha \right)\left(x^{2}-(2-\alpha )x+1+{\sqrt {7}}-\alpha \right),}
with {\displaystyle \alpha ={\sqrt {4+2{\sqrt {7}}}}.}{\displaystyle \alpha ={\sqrt {4+2{\sqrt {7}}}}.} Also, Euler pointed out that

{\displaystyle x^{4}+a^{4}=\left(x^{2}+a{\sqrt {2}}\cdot x+a^{2}\right)\left(x^{2}-a{\sqrt {2}}\cdot x+a^{2}\right).}{\displaystyle x^{4}+a^{4}=\left(x^{2}+a{\sqrt {2}}\cdot x+a^{2}\right)\left(x^{2}-a{\sqrt {2}}\cdot x+a^{2}\right).}
A first attempt at proving the theorem was made by d’Alembert in 1746, but his proof was incomplete. Among other problems, it assumed implicitly a theorem (now known as Puiseux’s theorem), which would not be proved until more than a century later and using the fundamental theorem of algebra. Other attempts were made by Euler (1749), de Foncenex (1759), Lagrange (1772), and Laplace (1795). These last four attempts assumed implicitly Girard’s assertion; to be more precise, the existence of solutions was assumed and all that remained to be proved was that their form was a + bi for some real numbers a and b. In modern terms, Euler, de Foncenex, Lagrange, and Laplace were assuming the existence of a splitting field of the polynomial p(z).

At the end of the 18th century, two new proofs were published which did not assume the existence of roots, but neither of which was complete. One of them, due to James Wood and mainly algebraic, was published in 1798 and it was totally ignored. Wood’s proof had an algebraic gap.[4] The other one was published by Gauss in 1799 and it was mainly geometric, but it had a topological gap, only filled by Alexander Ostrowski in 1920, as discussed in Smale (1981).[5] The first rigorous proof was published by Argand in 1806 (and revisited in 1813);[6] it was also here that, for the first time, the fundamental theorem of algebra was stated for polynomials with complex coefficients, rather than just real coefficients. Gauss produced two other proofs in 1816 and another incomplete version of his original proof in 1849.

The first textbook containing a proof of the theorem was Cauchy’s Cours d’analyse de l’École Royale Polytechnique (1821). It contained Argand’s proof, although Argand is not credited for it.

None of the proofs mentioned so far is constructive. It was Weierstrass who raised for the first time, in the middle of the 19th century, the problem of finding a constructive proof of the fundamental theorem of algebra. He presented his solution, which amounts in modern terms to a combination of the Durand–Kerner method with the homotopy continuation principle, in 1891. Another proof of this kind was obtained by Hellmuth Kneser in 1940 and simplified by his son Martin Kneser in 1981.

Without using countable choice, it is not possible to constructively prove the fundamental theorem of algebra for complex numbers based on the Dedekind real numbers (which are not constructively equivalent to the Cauchy real numbers without countable choice).[7] However, Fred Richman proved a reformulated version of the theorem that does work.[8]

Proofs
All proofs below involve some mathematical analysis, or at least the topological concept of continuity of real or complex functions. Some also use differentiable or even analytic functions. This fact has led to the remark that the Fundamental Theorem of Algebra is neither fundamental, nor a theorem of algebra.[citation needed]

Some proofs of the theorem only prove that any non-constant polynomial with real coefficients has some complex root. This is enough to establish the theorem in the general case because, given a non-constant polynomial p(z) with complex coefficients, the polynomial

{\displaystyle q(z)=p(z){\overline {p({\overline {z}})}}}q(z)=p(z){\overline {p({\overline {z}})}}
has only real coefficients and, if z is a zero of q(z), then either z or its conjugate is a root of p(z).

A large number of non-algebraic proofs of the theorem use the fact (sometimes called “growth lemma”) that an n-th degree polynomial function p(z) whose dominant coefficient is 1 behaves like zn when |z| is large enough. A more precise statement is: there is some positive real number R such that:

{\displaystyle {\tfrac {1}{2}}|z^{n}|<|p(z)|<{\tfrac {3}{2}}|z^{n}|}{\tfrac {1}{2}}|z^{n}|<|p(z)|<{\tfrac {3}{2}}|z^{n}|
when |z| > R.

Complex-analytic proofs
Find a closed disk D of radius r centered at the origin such that |p(z)| > |p(0)| whenever |z| ≥ r. The minimum of |p(z)| on D, which must exist since D is compact, is therefore achieved at some point z0 in the interior of D, but not at any point of its boundary. The Maximum modulus principle (applied to 1/p(z)) implies then that p(z0) = 0. In other words, z0 is a zero of p(z).

A variation of this proof does not require the use of the maximum modulus principle (in fact, the same argument with minor changes also gives a proof of the maximum modulus principle for holomorphic functions). If we assume by contradiction that a := p(z0) ≠ 0, then, expanding p(z) in powers of z − z0 we can write

{\displaystyle p(z)=a+c_{k}(z-z_{0})^{k}+c_{k+1}(z-z_{0})^{k+1}+\cdots +c_{n}(z-z_{0})^{n}.}{\displaystyle p(z)=a+c_{k}(z-z_{0})^{k}+c_{k+1}(z-z_{0})^{k+1}+\cdots +c_{n}(z-z_{0})^{n}.}
Here, the cj are simply the coefficients of the polynomial z → p(z + z0), and we let k be the index of the first coefficient following the constant term that is non-zero. But now we see that for z sufficiently close to z0 this has behavior asymptotically similar to the simpler polynomial {\displaystyle q(z)=a+c_{k}(z-z_{0})^{k}}q(z)=a+c_{k}(z-z_{0})^{k},

in the sense that (as is easy to check) the function

{\displaystyle \left|{\frac {p(z)-q(z)}{(z-z_{0})^{k+1}}}\right|}\left|{\frac {p(z)-q(z)}{(z-z_{0})^{k+1}}}\right|
is bounded by some positive constant M in some neighborhood of z0. Therefore, if we define {\displaystyle \theta _{0}=(\arg(a)+\pi -\arg(c_{k}))/k}\theta _{0}=(\arg(a)+\pi -\arg(c_{k}))/k and let {\displaystyle z=z_{0}+re^{i\theta _{0}}}z=z_{0}+re^{i\theta _{0}}, then for any sufficiently small positive number r (so that the bound M mentioned above holds), using the triangle inequality we see that

https://vexare4629.medium.com/martin-morocky-born-31-march-1983-known-by-his-902c97847c76
https://arrabble.medium.com/mshasho-clothing-is-a-namibian-clothing-brand-founded-db8543163031
https://arrabble.medium.com/martin-morocky-born-31-march-1983-known-by-his-stage-name-as-the-dogg-646979461508
https://vexare4629.medium.com/namibian-music-awards-441b2392403b
https://vexare4629.medium.com/namibia-n%C9%99%CB%88m%C9%AAbi%C9%99-about-this-soundlisten-n%C3%A6%CB%88-16-17-officially-the-republic-of-namibia-e21df090d8c8
https://arrabble.medium.com/an-electromagnetic-field-also-em-field-is-a-classical-i-e-non-quantum-6fc1a608d934
https://jacap31175.medium.com/in-nuclear-physics-and-particle-physics-the-weak-interaction-d752c8a72d9
https://dedecab107.medium.com/classical-electromagnetism-or-classical-electrodynamics-c9d3b3bb0533
https://jacap31175.medium.com/particle-physics-cp-violation-is-a-violation-of-cp-symmetry-db0cb19c7840
https://dedecab107.medium.com/maxwells-equations-are-a-set-of-coupled-partial-differential-equations-that-together-108fab7e826c
https://jacap31175.medium.com/the-mathematical-concept-of-a-hilbert-space-named-after-david-hilbert-generalizes-4093c9b3b383
https://dedecab107.medium.com/particle-physics-a-magnetic-monopole-is-a-hypothetical-elementary-particle-38240eff467f
https://yobare1608.medium.com/mathematics-a-complex-number-is-a-number-that-can-be-expressed-f7d1a97b971f
https://mafor48407.medium.com/euclidean-space-is-the-fundamental-space-of-classical-geometry-ef959167204e

The fundamental theorem of algebra states that every non-constant

Written by Yobare