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Encyclopedia > Fundamental theorem of algebra

In mathematics, the fundamental theorem of algebra states that every complex polynomial p(z) in one variable and of degree n ≥ 1 has some complex root. In other words, the field of complex numbers is algebraically closed and therefore, as for any other algebraically closed field, the equation p(z) = 0 has n roots (not necessarily distinct). Euclid, detail from The School of Athens by Raphael. ... In mathematics, there are a number of fundamental theorems for different fields. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... The degree of a term of a polynomial in one variable is the exponent on the variable in that term; the degree of a polynomial is the maximum of the degrees of all terms in the polynomial. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... This article presents the essential definitions. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ...

The name of the theorem is now considered something of a misnomer by many mathematicians, since it is not fundamental for contemporary algebra. Look up Misnomer in Wiktionary, the free dictionary. ... Linear algebra lecture at Helsinki University of Technology This article is about the branch of mathematics; for other uses of the term see algebra (disambiguation). ...



Peter Rothe (Petrus Roth), in his book Arithmetica Philosophica (published in 1608), 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 complex 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 x4 = 4x − 3, although incomplete, has four solutions: 1, 1, − 1 + i2, and − 1 − i2. Albert Girard (1595–1632) was a French-born mathematician. ...

As will be mentioned again below, it follows from the fundamental theorem of algebra that every polynomial with real coefficients and degree greater than 0 can be written as a product of polynomials with real coefficients whose degree is either 1 or 2. However, in 1702 Leibniz 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 in which he was told that his polynomial happened to be equal to Gottfried Wilhelm Leibniz (also von Leibni(t)z)[1] (July 1 (June 21 Old Style) 1646, Leipzig – November 14, 1716, Hanover) was a German polymath, deemed a universal [1] genius in his day and since. ... Nicolaus I Bernoulli (21 October 1687, Basel, Switzerland – 29 November 1759, Basel) was a Swiss mathematician. ... Leonhard Euler by Emanuel Handmann. ...


where α is the square root of 4 + 27, whereas


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 be proved only more than a century later (and furthermore the proof assumed 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). Jean le Rond dAlembert, pastel by Maurice Quentin de la Tour Jean le Rond dAlembert (November 16, 1717 – October 29, 1783) was a French mathematician, mechanician, physicist and philosopher. ... In mathematics, a Puiseux expansion is a formal power series expansion of an algebraic function. ... Leonhard Euler by Emanuel Handmann. ... Joseph Louis Lagrange Joseph Louis Lagrange (January 25, 1736 – April 10, 1813; born Giuseppe Luigi Lagrangia in Turin, Lagrange moved to Paris (1787) and became a French citizen, adopting the French translation of his name, Joseph Louis Lagrange) was an Italian-French mathematician and astronomer who made important contributions to... Pierre-Simon Laplace. ... In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors X − ai, and such that the ai generate L over K. It can be shown that such splitting fields exist...

At the end of the 18th century two new proofs were published which did not assume the existence of roots. 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. The other one was published by Gauss in 1799 and it was mainly geometric, but it had a topological gap. A rigorous proof was published by Argand in 1806; it was 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 version of his original proof in 1849. ... Jean-Robert Argand was an accountant and bookkeeper in Paris who was only an amateur mathematician. ...

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. Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...

None of the proofs mentioned so far is constructive. It was Weierstrass who raised for the first time, in 1891, the problem of finding a constructive proof of the fundamental theorem of algebra. Such a proof was obtained by Hellmuth Kneser in 1940 and simplified by his son Martin Kneser in 1981. In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... Karl Theodor Wilhelm Weierstraß (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. (The letter ß may be transliterated as ss; one often writes Weierstrass. ... Hellmuth Kneser (April 16, 1898 - August 23, 1973) was a german mathematician. ...


All proofs of the fundamental theorem involve some analysis, or more precisely, the concept of continuity of real or complex functions. Some proofs also use differentiable or even analytic functions. Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the derivative is defined as the instantaneous rate of change of a function. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ...

Some proofs of the theorem only prove that any polynomial with real coefficients has some complex root. This is enough to establish the theorem in the general case because, given a polynomial p(z) with complex coefficients, the polynomial 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 p(z) behaves like zn when | z | is large enough. A more precise statement is: there is some positive real number R such that

| z | n / 2 < | p(z) | < 3 | z | n / 2

when | z | > R.

We mention approaches via complex analysis, topology, and algebra: Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Linear algebra lecture at Helsinki University of Technology This article is about the branch of mathematics; for other uses of the term see algebra (disambiguation). ...

Analytical 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 minimum modulus principle implies then that p(z0) = 0. In other words, z0 is a zero of p(z). In geometry, a disk is the region in a plane contained inside of a circle. ... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ... In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus |f | cannot exhibit a true local maximum within the domain of f. ...

Another analytical proof can be obtained along this line of thought observing that, since | p(z) | > | p(0) | outside D, the minimum of | p(z) | on the whole complex plane is achieved at z0. If | p(z0) | > 0, then 1 / p is a bounded holomorphic function in the entire complex plane since, for each complex number z, | 1 / p(z) || 1 / p(z0) | . Applying Liouville's theorem, which states that a bounded entire function must be constant, this would imply that 1 / p is constant and therefore that p is constant. This gives a contradiction, and hence p(z0) = 0 Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... Liouvilles theorem in complex analysis states that every bounded (i. ...

Topological proofs

As an alternative to the use of Liouville's theorem in the previous proof, we can write p(z) as a polynomial in zz0: there is some natural number k and there are some complex numbers ckck + 1, … , cn such that ck ≠ 0 and that

p(z) = p(z0) + ck(zz0)k + ck + 1(zz0)k + 1 +  ··· + cn(zz0)n.

It follows that if a is a kth root of p(z0) / ck and if t is positive and sufficiently small, then | p(z0 + ta) | < | p(z0) | , which is impossible, since | p(z0) | is the minimum of | p | on D.

For another topological proof by contradiction, assume p(z) has no zeros. Choose a large positive number R such that, for | z | = R, the leading term zn of p(z) dominates all other terms combined; in other words, such that | z | n > | an − 1zn − 1 +  ···  + a0 | . As z traverses the circle | z | = R once counter-clockwise, p(z), like zn, winds n times counter-clockwise around 0. At the other extreme, with | z | = 0, the “curve” p(z) is simply the single (nonzero) point p(0), whose winding number is clearly 0. If the loop followed by z is continuously deformed between these extremes, the path of p(z) also deforms continuously. Since p(z) has no zeros, the path can never cross over 0 as it deforms, and hence its winding number with respect to 0 will never change. However, given that the winding number started as n and ended as 0, this is absurd. Therefore, p(z) has at least one zero. A point z0 and a curve C In mathematics, the winding number is a topological invariant playing a leading role in complex analysis. ...

Algebraic proof

This proof uses only two facts about real numbers whose proof require analysis (more precisely, the intermediate value theorem), namely: In analysis, the intermediate value theorem is either of two theorems of which an account is given below. ...

  • every polynomial with odd degree and real coefficients has some real root;
  • every non-negative real number has a square root.

It follows from the second assertion that, if a and b are real numbers, then there are complex numbers z1 and z2 such that the polynomial z2 + az + b is equal to (zz1)(zz2).

As it was mentioned above, it suffices to check that the fundamental theorem is true for all polynomials p(z) with real coefficients. The theorem can be proved by induction on the greatest non-negative integer k such that 2k divides the degree n of p(z). Let F be a splitting field of p(z) (seen as a polynomial with complex coefficients); in other words, the field F contains C and there are elements z1z2, …, zn in F such that In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors X − ai, and such that the ai generate L over K. It can be shown that such splitting fields exist...

p(z) = (zz1)(zz2) ··· (zzn).

If k = 0, then n is odd, and therefore p(z) has a real root. Now, suppose that n = 2km (with m odd and k > 0) and that the theorem is already proved when the degree of the polynomial has the form 2k − 1m' with m' odd. For a real number t, define:

q_t(z)=prod_{1le i<jle n}left(z-z_i-z_j-tz_iz_jright).,

Then the coefficients of qt(z) are symmetric polynomials in the zi's with real coefficients. Therefore, they can be expressed as polynomials with real coefficients in the elementary symmetric polynomials, that is, in -a_1, a_2, dots, (-1)^na_n. So qt has in fact real coefficients. Furthermore, the degree of qt is n(n − 1) / 2 = 2k − 1m(n − 1), and m(n − 1) is an odd number. So, using the induction hypothesis, qt has, at least, one real root; in other words, zi + zj + tzizj is real for two distinct elements i and j from {1,dots,n}. Since there are more real numbers than pairs (i,j), one can find distinct real numbers t and s such that zi + zj + tzizj and zi + zj + szizj are real (for the same i and j). So, both zi + zj and zizj are real numbers, and therefore zi and zj are complex numbers, since they are roots of the polynomial z2 − (z1 + z2)z + z1z2. In mathematics, a symmetric polynomial is a polynomial in n variables , such that if some of the variables are interchanged, the polynomial stays the same. ... In mathematics, specifically in commutative algebra, elementary symmetric polynomials are the basic building blocks for symmetric polynomials, in the sense that every symmetric polynomial can be expressed as a sum of products of the elementary symmetric polynomials. ...


Since the fundamental theorem of algebra can be seen as the statement that the field of complex numbers is algebraically 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 about the relationship between the field of real numbers and the field of complex numbers: In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ...

  • The field of complex numbers is the algebraic closure of the field of real numbers.
  • Every polynomial in one variable x with real coefficients is the product of a constant, polynomials of the form x + a with a real, and polynomials of the form x2 + ax + b with a and b real and a2 − 4b < 0 (which is the same thing as saying that the polynomial x2 + ax + b has no real roots).
  • Every rational function in one variable x, with real coefficients, can be written as the sum of a polynomial function with rational functions of the form a / (xb)n (where n is a natural number, and a and b are real numbers), and rational functions of the form (ax + b) / (x2 + cx + d)n (where n is a natural number, and a, b, c, and d are real numbers such that c2 − 4d < 0). A corollary of this is that every rational function in one variable and real coefficients has an elementary primitive.
  • Every algebraic extension of the real field is isomorphic either to the real field or to the complex field.

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... A theorem is a statement which can be proven true within some logical framework. ... In differential algebra, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ − × ÷). The trigonometric functions and their inverses are assumed to be included in the elementary functions by using... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... In abstract algebra, a field extension L /K is called algebraic if every element of L is algebraic over K, i. ...


  • A.-L. Cauchy, Cours d'Analyse de l'École Royale Polytechnique, 1ère partie: Analyse Algébrique, 1992, Éditions Jacques Gabay, ISBN 2-87647-053-5
  • B. Fine and G. Rosenberger, The Fundamental Theorem of Algebra, 1997, Springer-Verlag, ISBN 0-387-94657-8
  • C. F. Gauss, “New Proof of the Theorem That Every Algebraic Rational Integral Function In One Variable can be Resolved into Real Factors of the First or the Second Degree”, 1799
  • C. Gilain, “Sur l'histoire du théorème fondamental de l'algèbre: théorie des équations et calcul intégral”, Archive for History of Exact Sciences, 42 (1991), 91–136
  • E. Netto and R. Le Vavasseur, “Les fonctions rationnelles §80–88: Le théorème fondamental”, in Encyclopédie des Sciences Mathématiques Pures et Appliquées, tome I, vol. 2, 1992, Éditions Jacques Gabay, ISBN 2-87647-101-9
  • R. Remmert, “The Fundamental Theorem of Algebra”, in Numbers, 1991, Springer-Verlag, ISBN 0-387-97497-0
  • D. E. Smith, “A Source Book in Mathematics”, 1959, Dover Publications, ISBN 0-486-64690-4
  • F. Smithies, “A forgotten paper on the fundamental theorem of algebra”, Notes & Records of the Royal Society, 54 (2000), 333–341
  • M. Spivak, Calculus, 1994, Publish or Perish, ISBN 0-914098-89-6
  • B. L. van der Waerden, Algebra I, 1991, Springer-Verlag, ISBN 0-387-97424-5

External Links

  • Proof of the Fundamental Theorem of Algebra, Fermat's Last Theorem Blog

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