Thefundamental theorem of algebra, also calledd'Alembert's theorem[1] or thed'Alembert–Gauss theorem,[2] states that every non-constant single-variablepolynomial withcomplexcoefficients has at least one complexroot. This includes polynomials with real coefficients, since every real number is a complex number with itsimaginary part equal to zero.
The theorem is also stated as follows: every non-zero, single-variable,degreen polynomial with complex coefficients has, counted withmultiplicity, exactlyn complex roots. The equivalence of the two statements can be proven through the use of successivepolynomial division.
Despite its name, it is not fundamental formodern algebra; it was named when algebra wassynonymous with thetheory of equations. Additionally, the theorem can not be proven in Algebra, requiring higher level mathematics to prove the theorem.
Peter Roth [de], in his bookArithmetica Philosophica (published in 1608, at Nürnberg, by Johann Lantzenberger),[3] wrote that a polynomial equation of degreen (with real coefficients)may haven solutions.Albert Girard, in his bookL'invention nouvelle en l'Algèbre (published in 1629), asserted that a polynomial equation of degreen hasn 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 1702Leibniz erroneously said that no polynomial of the typex4 +a4 (witha real and distinct from 0) can be written in such a way. Later,Nikolaus Bernoulli made the same assertion concerning the polynomialx4 − 4x3 + 2x2 + 4x + 4, but he got a letter fromEuler in 1742[4] in which it was shown that this polynomial is equal to
withEuler also pointed out that
A first attempt at proving the theorem was made byd'Alembert in 1746, but his proof was incomplete. Among other problems, it assumed implicitly a theorem (now known asPuiseux's theorem), which would not be proved until more than a century later and using the fundamental theorem of algebra. Other attempts were made byEuler (1749),de Foncenex (1759),Lagrange (1772), andLaplace (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 wasa + bi for some real numbersa andb. In modern terms, Euler, de Foncenex, Lagrange, and Laplace were assuming the existence of asplitting field of the polynomialp(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 toJames Wood and mainly algebraic, was published in 1798 and it was totally ignored. Wood's proof had an algebraic gap.[5] The other one was published byGauss in 1799 and it was mainly geometric, but it had a topological gap, only filled byAlexander Ostrowski in 1920, as discussed in Smale (1981).[6]
The first rigorous proof was published byArgand, anamateur mathematician, in 1806 (and revisited in 1813);[7] 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.
None of the proofs mentioned so far isconstructive. It wasWeierstrass who raised for the first time, in the middle of the 19th century, the problem of finding aconstructive proof of the fundamental theorem of algebra. He presented his solution, which amounts in modern terms to a combination of theDurand–Kerner method with thehomotopy continuation principle, in 1891. Another proof of this kind was obtained byHellmuth Kneser in 1940 and simplified by his sonMartin Kneser in 1981.
Without usingcountable choice, it is not possible to constructively prove the fundamental theorem of algebra for complex numbers based on theDedekind real numbers (which are not constructively equivalent to the Cauchy real numbers without countable choice).[8] However,Fred Richman proved a reformulated version of the theorem that does work.[9]
Every univariate polynomial of positive degree with complex coefficients has at least one complexroot.
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 itscomplex 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 degreen with complex coefficients can befactorized as where are complex numbers.
Then complex numbers are the roots of the polynomial. If a root appears in several factors, it is amultiple root, and the number of its occurrences is, by definition, themultiplicity of the root.
The proof that this statement results from the previous ones is done byrecursion onn: when a root has been found, thepolynomial 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, ifr 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 thecomplex conjugate root theorem). Conversely, if one has a factor of degree two, thequadratic 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 wherec is a real number and each is amonic 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.
All proofs below involve somemathematical analysis, or at least thetopological concept ofcontinuity of real or complex functions. Some also usedifferentiable or evenanalytic functions. This requirement has led to the remark that the Fundamental Theorem of Algebra is neither fundamental, nor a theorem of algebra.[10]
Some proofs of the theorem only prove that any non-constant polynomial withreal coefficients has some complex root. This lemma is enough to establish the general case because, given a non-constant polynomialp with complex coefficients, the polynomial
has only real coefficients, and, ifz is a root ofq, then eitherz or its conjugate is a root ofp. Here, is the polynomial obtained by replacing each coefficient ofp with itscomplex conjugate; the roots of are exactly the complex conjugates of the roots ofp.
Many non-algebraic proofs of the theorem use the fact (sometimes called the "growth lemma") that a polynomial functionp(z) of degreen whose dominant coefficient is 1 behaves likezn when |z| is large enough. More precisely, there is some positive real numberR such that
Even without using complex numbers, it is possible to show that a real-valued polynomialp(x):p(0) ≠ 0 of degreen > 2 can always be divided by some quadratic polynomial with real coefficients.[11] In other words, for some real-valueda andb, the coefficients of the linear remainder on dividingp(x) byx2 −ax −b simultaneously become zero.
whereq(x) is a polynomial of degreen − 2. The coefficientsRp(x)(a,b) andSp(x)(a,b) are independent ofx and completely defined by the coefficients ofp(x). In terms of representation,Rp(x)(a,b) andSp(x)(a,b) are bivariate polynomials ina andb. 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 ofb, all the roots of bothRp(x)(a,b) andSp(x)(a,b) in the variablea are real-valued and alternating each other (interlacing property). Utilizing aSturm-like chain that containRp(x)(a,b) andSp(x)(a,b) as consecutive terms, interlacing in the variablea can be shown for all consecutive pairs in the chain wheneverb has sufficiently large negative value. AsSp(a,b = 0) =p(0) has no roots, interlacing ofRp(x)(a,b) andSp(x)(a,b) in the variablea fails atb = 0. Topological arguments can be applied on the interlacing property to show that the locus of the roots ofRp(x)(a,b) andSp(x)(a,b) must intersect for some real-valueda andb < 0.
Find a closeddiskD of radiusr centered at the origin such that |p(z)| > |p(0)| whenever |z| ≥ r. The minimum of |p(z)| onD, which must exist sinceD iscompact, is therefore achieved at some pointz0 in the interior ofD, but not at any point of its boundary. Themaximum modulus principle applied to 1/p(z) implies thatp(z0) = 0. In other words,z0 is a zero ofp(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, ifa :=p(z0) ≠ 0, then, expandingp(z) in powers ofz −z0, we can write
Here, thecj are simply the coefficients of the polynomialz →p(z +z0) after expansion, andk is the index of the first non-zero coefficient following theconstant term. Forz sufficiently close toz0 this function has behavior asymptotically similar to the simpler polynomial. More precisely, the function
for some positive constantM in some neighborhood ofz0. Therefore, if we define and let tracing a circle of radiusr > 0 aroundz, then for any sufficiently smallr (so that the boundM holds), we see that
Whenr is sufficiently close to 0 this upper bound for |p(z)| is strictly smaller than |a|, contradicting the definition ofz0. Geometrically, we have found an explicit direction θ0 such that if one approachesz0 from that direction one can obtain valuesp(z) smaller in absolute value than |p(z0)|.
Anotheranalytic proof can be obtained along this line of thought observing that, since |p(z)| > |p(0)| outsideD, the minimum of |p(z)| on the whole complex plane is achieved atz0. If |p(z0)| > 0, then 1/p is a boundedholomorphic function in the entire complex plane since, for each complex numberz, |1/p(z)| ≤ |1/p(z0)|. ApplyingLiouville's theorem, which states that a bounded entire function must be constant, this would imply that 1/p is constant and therefore thatp is constant. This gives a contradiction, and hencep(z0) = 0.[12]
Yet another analytic proof uses theargument principle. LetR be a positive real number large enough so that every root ofp(z) has absolute value smaller thanR; such a number must exist because every non-constant polynomial function of degreen has at mostn zeros. For eachr > R, consider the number
wherec(r) is the circle centered at 0 with radiusr oriented counterclockwise; then theargument principle says that this number is the numberN of zeros ofp(z) in the open ball centered at 0 with radiusr, which, sincer > R, is the total number of zeros ofp(z). On the other hand, the integral ofn/z alongc(r) divided by 2πi is equal ton. But the difference between the two numbers is
The numerator of the rational expression being integrated has degree at mostn − 1 and the degree of the denominator isn + 1. Therefore, the number above tends to 0 asr → +∞. But the number is also equal toN − n and soN = n.
Another complex-analytic proof can be given by combininglinear algebra with theCauchy theorem. To establish that every complex polynomial of degreen > 0 has a zero, it suffices to show that every complexsquare matrix of sizen > 0 has a (complex)eigenvalue.[13] The proof of the latter statement isby contradiction.
LetA be a complex square matrix of sizen > 0 and letIn be the unit matrix of the same size. AssumeA has no eigenvalues. Consider theresolvent function
which is ameromorphic function on the complex plane with values in the vector space of matrices. The eigenvalues ofA are precisely the poles ofR(z). Since, by assumption,A has no eigenvalues, the functionR(z) is anentire function andCauchy theorem implies that
On the other hand,R(z) expanded as a geometric series gives:
This formula is valid outside the closeddisc of radius (theoperator norm ofA). Let Then
(in which only the summandk = 0 has a nonzero integral). This is a contradiction, and soA has an eigenvalue.
Finally,Rouché's theorem gives perhaps the shortest proof of the theorem.
Animation illustrating the proof on the polynomial
Suppose the minimum of |p(z)| on the whole complex plane is achieved atz0; it was seen at the proof which uses Liouville's theorem that such a number must exist. We can writep(z) as a polynomial inz − z0: there is some natural numberk and there are some complex numbersck,ck + 1, ...,cn such thatck ≠ 0 and:
Ifp(z0) is nonzero, it follows that ifa is akth root of −p(z0)/ck and ift is positive and sufficiently small, then |p(z0 + ta)| < |p(z0)|, which is impossible, since |p(z0)| is the minimum of |p| onD.
For another topological proof by contradiction, suppose that the polynomialp(z) has no roots, and consequently is never equal to 0. Think of the polynomial as a map from the complex plane into the complex plane. It maps any circle |z| = R into a closed loop, a curveP(R). We will consider what happens to thewinding number ofP(R) at the extremes whenR is very large and whenR = 0. WhenR is a sufficiently large number, then the leading termzn ofp(z) dominates all other terms combined; in other words,
Whenz traverses the circle once counter-clockwise then windsn times counter-clockwise around the origin (0,0), andP(R) likewise. At the other extreme, with |z| = 0, the curveP(0) is merely the single pointp(0), which must be nonzero becausep(z) is never zero. Thusp(0) must be distinct from the origin (0,0), which denotes 0 in the complex plane. The winding number ofP(0) around the origin (0,0) is thus 0. Now changingR continuously willdeform the loop continuously. At someR the winding number must change. But that can only happen if the curveP(R) includes the origin (0,0) for someR. But then for somez on that circle |z| = R we havep(z) = 0, contradicting our original assumption. Therefore,p(z) has at least one zero.
These proofs of the Fundamental Theorem of Algebra must make use of the following two facts about real numbers that are not algebraic but require only a small amount of analysis (more precisely, theintermediate value theorem in both cases):
every polynomial with an odd degree and real coefficients has some real root;
every non-negative real number has a square root.
The second fact, together with thequadratic formula, implies the theorem for real quadratic polynomials. In other words, algebraic proofs of the fundamental theorem actually show that ifR is anyreal-closed field, then its extensionC =R(√−1) is algebraically closed.
As mentioned above, it suffices to check the statement "every non-constant polynomialp(z) with real coefficients has a complex root". This statement can be proved by induction on the greatest non-negative integerk such that 2k divides the degreen ofp(z). Leta be the coefficient ofzn inp(z) and letF be asplitting field ofp(z) overC; in other words, the fieldF containsC and there are elementsz1,z2, ...,zn inF such that
Ifk = 0, thenn is odd, and thereforep(z) has a real root. Now, suppose thatn = 2km (withm odd andk > 0) and that the theorem is already proved when the degree of the polynomial has the form 2k − 1m′ withm′ odd. For a real numbert, define:
Then the coefficients ofqt(z) aresymmetric polynomials in thezi with real coefficients. Therefore, they can be expressed as polynomials with real coefficients in theelementary symmetric polynomials, that is, in −a1,a2, ..., (−1)nan. Soqt(z) has in factreal coefficients. Furthermore, the degree ofqt(z) isn(n − 1)/2 = 2k−1m(n − 1), andm(n − 1) is an odd number. So, using the induction hypothesis,qt has at least one complex root; in other words,zi + zj + tzizj is complex for two distinct elementsi andj from {1, ...,n}. Since there are more real numbers than pairs (i,j), one can find distinct real numberst ands such thatzi + zj + tzizj andzi + zj + szizj are complex (for the samei andj). So, bothzi + zj andzizj are complex numbers. It is easy to check that every complex number has a complex square root, thus every complex polynomial of degree 2 has a complex root by the quadratic formula. It follows thatzi andzj are complex numbers, since they are roots of the quadratic polynomialz2 − (zi + zj)z + zizj.
Joseph Shipman showed in 2007 that the assumption that odd degree polynomials have roots is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed (so "odd" can be replaced by "odd prime" and this holds for fields of all characteristics).[14] For axiomatization of algebraically closed fields, this is the best possible, as there are counterexamples if a single prime is excluded. However, these counterexamples rely on −1 having a square root. If we take a field where −1 has no square root, and every polynomial of degreen ∈ I has a root, whereI is any fixed infinite set of odd numbers, then every polynomialf(x) of odd degree has a root (since(x2 + 1)kf(x) has a root, wherek is chosen so thatdeg(f) + 2k ∈I).
Another algebraic proof of the fundamental theorem can be given usingGalois theory. It suffices to show thatC has no proper finitefield extension.[15] LetK/C be a finite extension. Since thenormal closure ofK overR still has a finite degree overC (orR), we may assumewithout loss of generality thatK is anormal extension ofR (hence it is aGalois extension, as every algebraic extension of a field ofcharacteristic 0 isseparable). LetG be theGalois group of this extension, and letH be aSylow 2-subgroup ofG, so that theorder ofH is a power of 2, and theindex ofH inG is odd. By thefundamental theorem of Galois theory, there exists a subextensionL ofK/R such that Gal(K/L) = H. As [L:R] = [G:H] is odd, and there are no nonlinear irreducible real polynomials of odd degree, we must haveL =R, thus [K:R] and [K:C] are powers of 2. Assuming by way of contradiction that [K:C] > 1, we conclude that the2-group Gal(K/C) contains a subgroup of index 2, so there exists a subextensionM ofC of degree 2. However,C has no extension of degree 2, because every quadratic complex polynomial has a complex root, as mentioned above. This shows that [K:C] = 1, and thereforeK =C, which completes the proof.
There exists still another way to approach the fundamental theorem of algebra, due to J. M. Almira and A. Romero: byRiemannian geometric arguments. The main idea here is to prove that the existence of a non-constant polynomialp(z) without zeros implies the existence of aflat Riemannian metric over the sphereS2. This leads to a contradiction since the sphere is not flat.
A Riemannian surface (M,g) is said to be flat if itsGaussian curvature, which we denote byKg, is identically null. Now, theGauss–Bonnet theorem, when applied to the sphereS2, claims that
which proves that the sphere is not flat.
Let us now assume thatn > 0 and
for each complex numberz. Let us define
Obviously,p*(z) ≠ 0 for allz inC. Consider the polynomialf(z) = p(z)p*(z). Thenf(z) ≠ 0 for eachz inC. Furthermore,
We can use this functional equation to prove thatg, given by
forw inC, and
forw ∈ S2\{0}, is a well defined Riemannian metric over the sphereS2 (which we identify with the extended complex planeC ∪ {∞}).
Now, a simple computation shows that
since the real part of an analytic function is harmonic. This proves thatKg = 0.
Since the fundamental theorem of algebra can be seen as the statement that the field of complex numbers isalgebraically closed, it follows that any theorem concerning algebraically closed fields applies to the field of complex numbers. Here are a few more consequences of the theorem, which are either about the field of real numbers or the relationship between the field of real numbers and the field of complex numbers:
The field of complex numbers is thealgebraic closure of the field of real numbers.
Every polynomial in one variablez with complex coefficients is the product of a complex constant and polynomials of the formz + a witha complex.
Every polynomial in one variablex with real coefficients can be uniquely written as the product of a constant, polynomials of the formx + a witha real, and polynomials of the formx2 + ax + b witha andb real anda2 − 4b < 0 (which is the same thing as saying that the polynomialx2 + ax + b has no real roots). (By theAbel–Ruffini theorem, the real numbersa andb are not necessarily expressible in terms of the coefficients of the polynomial, the basic arithmetic operations and the extraction ofn-th roots.) This implies that the number of non-real complex roots is always even and remains even when counted with their multiplicity.
Everyrational function in one variablex, with real coefficients, can be written as the sum of a polynomial function with rational functions of the forma/(x − b)n (wheren is anatural number, anda andb are real numbers), and rational functions of the form (ax + b)/(x2 + cx + d)n (wheren is a natural number, anda,b,c, andd are real numbers such thatc2 − 4d < 0). Acorollary of this is that every rational function in one variable and real coefficients has anelementaryprimitive.
Everyalgebraic extension of the real field is isomorphic either to the real field or to the complex field.
While the fundamental theorem of algebra states a general existence result, it is of some interest, both from the theoretical and from the practical point of view, to have information on the location of the zeros of a given polynomial. The simplest result in this direction is a bound on the modulus: all zeros ζ of a monic polynomial satisfy an inequality |ζ| ≤R∞, where
As stated, this is not yet an existence result but rather an example of what is called ana priori bound: it says thatif there are solutions then they lie inside the closed disk of center the origin and radiusR∞. However, once coupled with the fundamental theorem of algebra it says that the disk contains in fact at least one solution. More generally, a bound can be given directly in terms of anyp-norm of then-vector of coefficients that is |ζ| ≤Rp, whereRp is precisely theq-norm of the 2-vectorq being the conjugate exponent ofp, for any 1 ≤p ≤ ∞. Thus, the modulus of any solution is also bounded by
for 1 <p < ∞, and in particular
(where we definean to mean 1, which is reasonable since 1 is indeed then-th coefficient of our polynomial). The case of a generic polynomial of degreen,
is of course reduced to the case of a monic, dividing all coefficients byan ≠ 0. Also, in case that 0 is not a root, i.e.a0 ≠ 0, bounds from below on the roots ζ follow immediately as bounds from above on, that is, the roots of
Finally, the distance from the roots ζ to any point can be estimated from below and above, seeing as zeros of the polynomial, whose coefficients are theTaylor expansion ofP(z) at
Let ζ be a root of the polynomial
in order to prove the inequality |ζ| ≤Rp we can assume, of course, |ζ| > 1. Writing the equation as
^See sectionLe rôle d'Euler in C. Gilain's articleSur l'histoire du théorème fondamental de l'algèbre: théorie des équations et calcul intégral.
^Concerning Wood's proof, see the articleA forgotten paper on the fundamental theorem of algebra, by Frank Smithies.
^Smale, Steve (1981),"The fundamental theorem of algebra and complexity theory"(PDF),Bulletin of the American Mathematical Society,4 (1):1–36,doi:10.1090/S0273-0979-1981-14858-8, retrieved2025-09-12 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..."
^For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman; 1998;A weak countable choice principle; available from[1]Archived 2020-02-19 at theWayback Machine.
^See Fred Richman; 1998;The fundamental theorem of algebra: a constructive development without choice; available from[2]Archived 2020-02-19 at theWayback Machine.
Gauss, Carl Friedrich (1799),Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse,Helmstedt: C. G. Fleckeisen (tr. New proof of the theorem that every integral rationalalgebraic function of one variable can be resolved into real factors of the first or second degree).
Gauss, Carl Friedrich (1866),Carl Friedrich Gauss Werke, vol. Band III, Königlichen Gesellschaft der Wissenschaften zu Göttingen
Weierstraß, Karl (1891), "Neuer Beweis des Satzes, dass jede ganze rationale Function einer Veränderlichen dargestellt werden kann als ein Product aus linearen Functionen derselben Veränderlichen",Sitzungsberichte der königlich preussischen Akademie der Wissenschaften zu Berlin, pp. 1085–1101 (tr. New proof of the theorem that every integral rational function of one variable can be represented as a product of linear functions of the same variable).
Gilain, Christian (1991), "Sur l'histoire du théorème fondamental de l'algèbre: théorie des équations et calcul intégral",Archive for History of Exact Sciences, vol. 42, no. 2, pp. 91–136,doi:10.1007/BF00496870,ISSN0003-9519,S2CID121468210 (tr. On the history of the fundamental theorem of algebra:theory of equations andintegral calculus.)
Netto, Eugen; Le Vavasseur, Raymond (1916), "Les fonctions rationnelles §80–88: Le théorème fondamental", in Meyer, François; Molk, Jules (eds.),Encyclopédie des Sciences Mathématiques Pures et Appliquées, tome I, vol. 2, Éditions Jacques Gabay (published 1992).ISBN978-2-87647-101-6. (tr. The rational functions §80–88: the fundamental theorem).
