Fundamental theorem of algebra

The ( Gaussian d'Alembertsche ) fundamental theorem of algebra states that every non-constant polynomial has at least one zero in the field of complex numbers. The coefficients of the polynomial can be arbitrary complex numbers - so that polynomials with integer or real coefficients are particularly possible.

Turning to the sentence, for example, on the polynomial, so it follows that the unsolvable in the field of real numbers equation in the field of complex numbers have at least one solution must.

The Fundamental Theorem of Algebra says that the complex numbers are algebraically closed.

• 4.1 Pure analytical proof
• 4.2 Proof methods with the topology
• 4.3 Proof of the intermediate value theorem and algebraic methods
• 4.4 Proof by methods of function theory 4.4.1 Proof of the theorem of Liouville
• 4.4.2 proof directly by the Cauchy integral theorem

Set

Be

A non-constant polynomial of degree, with complex coefficients. Then the polynomial has a complex zero, ie there is a number such that the following holds. More even considered that the number of zeros if they are counted with proper multiplicity, a total equal to the degree of the polynomial is.

Polynomials with real coefficients

If P is a polynomial over the real numbers, if so are all of the coefficients, the corresponding zeros are not necessarily real. It applies: If a non- real zero of P, as well as their complex conjugate is a root of P. If a multiple zero of P, so has the same multiplicity. In factorized notation of the polynomial can therefore be the associated linear factors always combine them to form a square factor. Multiplied out, this second-order polynomial again purely real coefficients:

It follows, conversely, that every real polynomial can be decomposed into real Polynomfaktoren of degree one or two. In this form of sentence in 1799 by Carl Friedrich Gauss was formulated as part of his doctoral thesis, which is already in its Latin title demonstrations nova theorematis omnem functionem algebraicam rational INTERGRAM unius variabilis in factores real primi vel secundi gradus resolvi posse announced this result ( German: New evidence of the theorem that every integral rational algebraic function can be decomposed into a variable in real factors first or second degree. )

Example

The polynomial equation

Has the solutions

The course, the zeros of the polynomial are. The solution 0 is thereby counted twice, as is apparent from the factorization of the polynomial:

We also used the speech "0 occurs with multiplicity 2 on " all other zeros occur with multiplicity 1. This example also shows that the nulls in general not (all) are real, even if the polynomial has real coefficients. Not Real zeros of polynomials with real coefficients occur but always in pairs complex conjugate to ( in our example).

Comments

From a polynomial can be split off of a zero with associated linear factor. ( For this example, the Horner - Ruffini method can be used. ) The spin-off is a reduced by one in degree polynomial for which you can repeat the process yields. Therefore, any non-constant polynomial decays via completely into a product of linear factors:

Which are the zeros of the polynomial.

Evidence

First formulations of the fundamental theorem can be found in the 17th century (Peter Roth, Albert Girard, René Descartes). The first published proof of Jean d' Alembert in 1746 was correct from the idea, but it contained loopholes that could be closed only with the methods of analysis of the 19th century. A simplified and also according to modern criteria still correct version of this proof was given by Jean -Robert Argand 1806.

The first complete proof of the fundamental theorem of algebra was given in 1799 by Carl Friedrich Gauss in his doctoral thesis. This proof also contains some analytical weaknesses that could be eliminated later. The second proof, which was introduced by Gauss in 1815 and published a year later, is based on ideas of Leonhard Euler and used as an analytical basis, unproven and without any evidence of need has been seen, only the intermediate value theorem of real analysis, or more precisely the special case that every polynomial of odd degree always has a real root.

A proof that at the same time an efficient calculation method includes, was published in 1859 ( and again in 1891) by Karl Weierstrass. The procedure contained in it is now called Durand- Kerner method.

We now know that several very different evidence, the concepts and ideas from analysis, algebra or topology include. The shortest of the Fundamental Theorem of Algebra by Cauchy and Liouville can be proved by methods of function theory.

In the following it is always a non-constant polynomial with complex coefficients, and in particular. This was regarded as a function.

Pure analytical proof

This proof was proposed in 1746 by d' Alembert, but only in 1806 by J.-R. Argand completed. The central message of this proof is that at any point that is not zero, a point z w in the environment can be specified, which results in a reduction in the amount of the function value. If the amount of the function values ​​so a minimum point, it must be a zero. Since the set is compact, and the amount associated with f is continuous, there is always such a minimum point and thus a zero.

For the central statement can develop f in z, that is,

If so z is a zero. Otherwise, choose the one k> 0 for which is considered the first and consider the two inequalities for

Both inequalities are satisfied for, and there is a finite, largest, so that they are satisfied on the whole interval. For a s of this interval with and so selective that the relationship is a real factor. Now applies to the interest amount of the function value by triangle inequality

Proof methods with the topology

A proof of this method was given in 1799 by Gauss. He divided the polynomial into real and imaginary parts. The zero sets of u and v are composed of individual one-dimensional arcs that connect a finite number of nodes in the plane. From each node comes from an even number of arcs. In no case may simply end an arc at one point. On each circle with sufficiently large radius, there are 2n zeros of u and 2n zeros of v, which alternate. Each contiguous part of the zeros of the graph of u on a big circle an even number of interfaces that include an odd number of interfaces of the zeros of the graph of v. This must protrude from u an arc of the graph of v from the contiguous portion of the graph. This is only possible if the graph of u and v intersect, but the cut point is a zero of f (z).

Modern versions of this proof using the notion of winding number. It is assumed that the polynomial f (z) does not possess complex zeros. Then for every s > 0 is a closed, continuous curve

Be constructed that the ( scaled ) Functional values ​​of the polynomial on the circle having a radius S passes. Since no function value is zero, a number of turns can be defined. Since the curve s changes continuously changing the parameter, the number of turns can only change when the changing curve crosses the zero point. By assumption the function f ( z) has no zeros, such a crossing of the zero point is not possible. Therefore, the number of turns for all s must be> 0 the same.

For very large values ​​of S, the graph of the curve corresponding to the nth power, or more precisely of the polynomial, more similar to, the number of turns N must therefore be constant. For very small values ​​of s, the curve of constant curve value is always similar, so you have the - for all s > 0 constant - at the same time have the value 0 turns. This is also only possible if n = 0 applies, that is, the polynomial is constant. For polynomials of higher degree, this argument leads to a contradiction, so it must be zeros z with f ( z) = 0 give.

Proof of the intermediate value theorem and algebraic methods

Such evidence was presented in 1815 by Gauss. It is used by the intermediate value theorem that every real polynomial of odd degree has at least one zero, and that quadratic equations with complex coefficients, elemental are detachable. The proof is as complete induction on the potency of the factor in the degree of the polynomial.

Assume first square-free and with real coefficients. The degree has a factorization with odd. The proof is as complete induction on the potency of the factor in the degree of the polynomial. If so there is a zero point by the intermediate value theorem. It is now provided in the induction step that all polynomials have degrees with odd with at least one zero.

It should, for the sake of simplicity, an ( abstract ) splitting field of the polynomial constructed in which it has the pairwise different (again abstract ) zeros,

In be the set of points is considered. Since the abstract zeros are pairwise different, are only a finite number of straight lines, which pass through at least two of these points, in particular a finite number of real increases such straight lines for which the difference takes the same value twice. For all other values ​​of the polynomial

Also square-free and symmetric in the abstract zeros. Therefore, the coefficients of polynomials can be the coefficients of and shown in, so for every real a polynomial with real coefficients and can be determined by means of resultants. The degree of is, k ( n-1) is an odd number. By the induction hypothesis, there are at least with a complex zero. Can from the partial derivatives to and within the zero complex numbers and are determined so that at least one of the zeros of a zero of is.

Also has real complex coefficients, so has only real coefficients. Each zero of the product is zero of a factor and thus to that yourself or as complex conjugate of a zero of. Is not now the square-free real polynomial, then with polynomial arithmetic ( Euclidean algorithm among others ) a factorization in ( non-constant ) square-free factors are found, each of which contains at least one zero.

Proof using methods of function theory

Proof of the theorem of Liouville

Because one exists, such that for all with. Because both the amount and therefore are continuous, and the disc is compact, there exists according to the Weierstrass point with a minimum sum of the function value for all. By construction even is a global minimum. If positive, the reciprocal function would be holomorphic on and limited by, so by the theorem of Liouville constant. Thus, f ( z) would be constant, which contradicts the assumption. So there is a zero ( in ).

Evidence directly by means of Cauchy's integral theorem

The fundamental theorem of algebra can be derived using elementary estimates even directly from the Cauchy integral theorem, namely as follows:

The polynomial can be represented in the form where another polynomial.

If we now assume, is without zero, it can be written as always:

Now it is for each of the path integral receives formed on reciprocal function on the Kreislinienweg and:

Due to the assumed zeros of freedom is

Holomorphic, which is also apparent as a result of Cauchy's integral theorem:

And from this:

This applies to any.

Now, however, and thus it follows from the last inequality directly:

Which is certainly wrong.

Thus, the assumed zeros of freedom is leads to a contradiction and must have a zero.

Evidence with methods of complex geometry

We take on as picture of the complex - projective space, that is,. The so- defined mapping of complex manifolds is holomorphic and thus open ( that is, the image of every open subset is open). Since is compact and continuous, the image is also compact, especially in completed. Thus, the picture is quite, because is connected. In particular, there is one which is displayed, i.e. a root of f

Proof using methods of differential topology

Similarly to the above proof of the complex geometry, we take on as a self-map of the sphere. So is ( real) differentiable and the set of critical points is as set of zeros of the derivative at last, so that the set of regular values ​​is connected. The cardinality of the preimage of a regular value is also locally constant as a function in ( is injective on environments of points in ). This shows that is surjective, for regular values ​​are thus always accepted and critical values ​​are assumed by definition.

Generalization of the fundamental theorem

The fundamental theorem of algebra can be generalized with the help of topological methods by applying the homotopy theory and the mapping degree continues:

It follows from the fundamental theorem by the leading coefficient, that takes a complex polynomial of degree as a constant.