The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, 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, it is not fundamental for modern algebra; it was named when algebra was synonymous with the theory of equations. Additionally, the theorem cannot be proven in Algebra: transcendental methods such as, for example, topology and the theory of holomorphic functions of one complex variable are required to prove the theorem.
History , in his book Arithmetica Philosophica (published in 1608, at Nürnberg, by Johann Lantzenberger),Rare books 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", where "incomplete" means that at least one 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 although incomplete, has four solutions (counting multiplicities): 1 (twice), and
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 (with real and distinct from 0) can be written in such a way. Later, Nikolaus Bernoulli made the same assertion concerning the polynomial , but he got a letter from Euler in 1742See section Le rôle d'Euler in C. Gilain's article Sur l'histoire du théorème fondamental de l'algèbre: théorie des équations et calcul intégral. in which it was shown that this polynomial is equal to
with Euler also pointed out that
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.Concerning Wood's proof, see the article A forgotten paper on the fundamental theorem of algebra, by Frank Smithies. 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). Smale writes "...I wish to point out what an immense gap Gauss's proof contained. It is a subtle point even today that a real algebraic plane curve cannot enter a disk without leaving. In fact, even though Gauss redid this proof 50 years later, the gap remained. It was not until 1920 that Gauss's proof was completed. In the reference Gauss, A. Ostrowski has a paper which does this and gives an excellent discussion of the problem as well..."
The first rigorous proof was published by Argand, an amateur mathematician, in 1806 (and revisited in 1813); 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).For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman; 1998; A weak countable choice principle; available from . However, Fred Richman proved a reformulated version of the theorem that does work.See Fred Richman; 1998; The fundamental theorem of algebra: a constructive development without choice; available from .
Equivalent statements There are several equivalent formulations of the theorem: Every univariate polynomial of positive degree with real coefficients has at least one complex root. Every univariate polynomial of positive degree with complex coefficients has at least one complex root. This implies immediately the previous assertion, as real numbers are also complex numbers. The converse results from the fact that one gets a polynomial with real coefficients by taking the product of a polynomial and its complex conjugate (obtained by replacing each coefficient with its complex conjugate). A root of this product is either a root of the given polynomial, or of its conjugate; in the latter case, the conjugate of this root is a root of the given polynomial. Every univariate polynomial of positive degree with complex coefficients can be factorized as where are complex numbers. The complex numbers are the roots of the polynomial. If a root appears in several factors, it is a multiple root, and the number of its occurrences is, by definition, the multiplicity of the root. The proof that this statement results from the previous ones is done by recursion on : when a root has been found, the polynomial division by provides a polynomial of degree whose roots are the other roots of the given polynomial. The next two statements are equivalent to the previous ones, although they do not involve any nonreal complex number. These statements can be proved from previous factorizations by remarking that, if is a non-real root of a polynomial with real coefficients, its complex conjugate is also a root, and is a polynomial of degree two with real coefficients (this is the complex conjugate root theorem). Conversely, if one has a factor of degree two, the quadratic formula gives a root. Every univariate polynomial with real coefficients of degree larger than two has a factor of degree two with real coefficients. Every univariate polynomial with real coefficients of positive degree can be factored as where is a real number and each is a monic polynomial of degree at most two with real coefficients. Moreover, one can suppose that the factors of degree two do not have any real root.
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 requirement has led to the remark that the Fundamental Theorem of Algebra is neither fundamental, nor a theorem of algebra.
Some proofs of the theorem only prove that any non-constant polynomial with real coefficients has some complex root. This lemma is enough to establish the general case because, given a non-constant polynomial with complex coefficients, the polynomial
has only real coefficients, and, if is a root of , then either or its conjugate is a root of . Here, is the polynomial obtained by replacing each coefficient of with its complex conjugate; the roots of are exactly the complex conjugates of the roots of .
Many non-algebraic proofs of the theorem use the fact (sometimes called the "growth lemma") that a polynomial function p(z) of degree n whose dominant coefficient is 1 behaves like zn when |z| is large enough. More precisely, there is some positive real number R such that
when |z| > R.
Real-analytic proofs Even without using complex numbers, it is possible to show that a real-valued polynomial p(x): p(0) ≠ 0 of degree n > 2 can always be divided by some quadratic polynomial with real coefficients. In other words, for some real-valued a and b, the coefficients of the linear remainder on dividing p(x) by x2 − ax − b simultaneously become zero.
where q(x) is a polynomial of degree n − 2. The coefficients Rp(x)(a, b) and Sp(x)(a, b) are independent of x and completely defined by the coefficients of p(x). In terms of representation, Rp(x)(a, b) and Sp(x)(a, b) are bivariate polynomials in a and b. In the flavor of Gauss's first (incomplete) proof of this theorem from 1799, the key is to show that for any sufficiently large negative value of b, all the roots of both Rp(x)(a, b) and Sp(x)(a, b) in the variable a are real-valued and alternating each other (interlacing property). Utilizing a Sturm-like chain that contain Rp(x)(a, b) and Sp(x)(a, b) as consecutive terms, interlacing in the variable a can be shown for all consecutive pairs in the chain whenever b has sufficiently large negative value. As Sp(a, b = 0) = p(0) has no roots, interlacing of Rp(x)(a, b) and Sp(x)(a, b) in the variable a fails at b = 0. Topological arguments can be applied on the interlacing property to show that the locus of the roots of Rp(x)(a, b) and Sp(x)(a, b) must intersect for some real-valued a and b < 0.
Sheldon Axler has a proof of the fundamental theorem of algebra using De Moivre's formula and extreme value theorem on a compact set in a textbook called Linear Algebra Done Right.
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 that p(z0) = 0. In other words, z0 is a zero of p(z).
A variation of this proof does not require the maximum modulus principle (in fact, a similar argument also gives a proof of the maximum modulus principle for holomorphic functions). Continuing from before the principle was invoked, if a := p(z0) ≠ 0, then, expanding p(z) in powers of z − z0, we can write