Fundamental theorem of algebra

The (Gauss-d'Alembert) fundamental theorem of algebra states that every non-constant polynomial in the range of complex numbers has at least one zero . The coefficients of the polynomial can be any complex numbers - in particular, polynomials with whole or real coefficients are included.

If one applies the theorem to the polynomial , for example , it follows that the equation that is unsolvable in the domain of real numbers must have at least one solution in the domain of complex numbers. ${\displaystyle z^{4}+15z^{2}+4}$ ${\displaystyle z^{4}+15z^{2}+4=0}$

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

Polynomials with real and complex coefficients

sentence

Be

${\displaystyle P(z)=\sum _{k=0}^{n}a_{k}\cdot z^{k}}$

a non-constant polynomial of degree , with complex coefficients . Then the polynomial has a complex zero, i.e. i.e. there is a number such that it holds. More precisely, it is particularly true that the number of zeros, if they are counted with the correct multiplicity, is overall equal to the degree of the polynomial. ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n\geq 1}$ ${\displaystyle a_{k}\in \mathbb {C} }$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle P(z)=0}$

Polynomials with real coefficients

Even if a polynomial over the real numbers, ie when all the coefficients in lying, its zeros are not necessarily real. But the following applies: If there is a non-real zero of , then its complex conjugate is also a zero of . Is a multiple zero of , then has the same multiplicity. In the factored notation of the polynomial, the associated linear factors can therefore always be combined into a quadratic factor . Once again, this second-degree polynomial has purely real coefficients: ${\displaystyle P}$ ${\displaystyle a_{k}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle w}$ ${\displaystyle P}$ ${\displaystyle {\bar {w}}}$ ${\displaystyle P}$ ${\displaystyle w}$ ${\displaystyle P}$ ${\displaystyle {\bar {w}}}$ ${\displaystyle (z-w)(z-{\bar {w}})}$

${\displaystyle (z-w)(z-{\bar {w}})=z^{2}-(w+{\bar {w}})z+w\cdot {\bar {w}}=z^{2}-2\operatorname {Re} (w)\cdot z+|w|^{2}}$

Conversely, it follows from this that every real polynomial can be broken down into real polynomial factors of degree one or two. In this form, the sentence was formulated in 1799 by Carl Friedrich Gauß as part of his doctoral thesis, which already announced this result in its Latin title Demonstratio nova theorematis omnem functionem algebraicam rational integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (German: New proof the theorem that every whole rational algebraic function in a variable can be broken down into real factors of the first or second degree. )

example

The polynomial equation

${\displaystyle P(x)=x^{5}-5x^{4}+17x^{3}-13x^{2}=0}$

has the solutions

${\displaystyle L=\{0^{(2)},1,2-3{\text{i}},2+3{\text{i}}\}}$ ,

which of course are the roots of the polynomial. The solution 0 is counted twice, as can be seen from the factorization of the polynomial :

${\displaystyle P(x)=x\cdot x\cdot (x-1)\cdot (x-2+3{\text{i}})\cdot (x-2-3{\text{i}})=x\cdot x\cdot (x-1)\cdot (x^{2}-4x+13)}$ .

One also uses the phrase “0 occurs with a multiplicity of 2”, all other zeros occur with a multiplicity of 1. This example also shows that the roots are generally not (all) real, even if the polynomial has real coefficients. However, non-real zeros of polynomials with real coefficients always appear in complex conjugate pairs (in the example above ). ${\displaystyle 2\pm 3i}$

Remarks

Of a polynomial can be of a zero with associated linear factor secede. (For example, the Horner-Ruffini method can be used for this.) The splitting off results in a polynomial reduced in degree by one, for which the procedure can be repeated. Therefore, every non-constant polynomial breaks down completely into a product of linear factors: ${\displaystyle f(z)}$ ${\displaystyle z_{0}}$ ${\displaystyle f(z_{0})=0}$ ${\displaystyle (z-z_{0})}$ ${\displaystyle \mathbb {C} }$

${\displaystyle f(z)=\sum _{k=0}^{n}a_{k}\cdot z^{k}=a_{n}\cdot \prod _{i=1}^{n}(z-z_{i})}$ ,

where they are the roots of the polynomial. ${\displaystyle z_{i}}$

proofs

The first formulations of the fundamental theorem can be found in the 17th century ( Peter Roth , Albert Girard , René Descartes ). Peter Roth (1608) suggested that -th degree equations have at most solutions, and Francois Viète gave examples of -th degree equations with the maximum number of solutions. Albert Girard was the first to suspect in 1629 ( L'invention en l'algèbre ) that there are always solutions, and already suspected complex solutions alongside real ones. Leonhard Euler gave a formulation of the fundamental theorem as complete factorization in the complex in today's sense. The first published proof by Jean d'Alembert in 1746 was theoretically correct, but it contained loopholes that could only be closed with the methods of analysis of the 19th century. A simplified version of this proof that was still correct according to modern criteria was given by Jean-Robert Argand in 1806. Other published attempts at proof come from Euler (1749), Joseph-Louis Lagrange (1772), building on Euler's proof, and Pierre Simon de Laplace (1795), who took a new approach using the discriminant of the polynomial. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$

The first complete proof of the fundamental theorem of algebra was given by Carl Friedrich Gauß as part of his dissertation in 1799 (and a note was made in his diary in October 1797). In contrast to his predecessors, Gauss also tackled the problem of proving the existence of roots in the complex and not tacitly assuming it. This proof also contains some analytical weaknesses that could only be eliminated later. The second proof, which was presented by Gauß in 1815 and published a year later, builds on the ideas of Leonhard Euler and uses as an analytical basis, unproven and without the need for proof, only the intermediate value theorem of real analysis, more precisely the special case that every polynomial of odd degree always has a real zero.

A proof, which at the same time includes an efficient calculation method, was published in 1859 (and again in 1891) by Karl Weierstrass . The process it contains is now known as the Durand-Kerner process .

We now know several very different proofs that contain terms and ideas from analysis , algebra or topology . The shortest possible way to prove the fundamental theorem of algebra according to Augustin-Louis Cauchy and Joseph Liouville is with methods of function theory .

In the following, always be a non-constant polynomial with complex coefficients and in particular . This is understood as a function . ${\displaystyle f(z)=a_{n}z^{n}+\dotsb +a_{1}z+a_{0}}$ ${\displaystyle a_{0},a_{n}\neq 0}$ ${\displaystyle f\colon \mathbb {C} \to \mathbb {C} }$

Purely analytical evidence

This proof was suggested by d'Alembert in 1746, but not until 1806 by J.-R. Argand completed. The central message of this evidence is that at any point that is not zero, a point in the area can be specified, which results in a reduction in the amount of the function value . So if the amount of the function values ​​has a minimum point, this must be a zero point. Since the set is compact and the amount is connected with continuous, there is always such a minimum point and thus a zero. ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle z+w}$ ${\displaystyle |f(z+w)|<|f(z)|}$ ${\displaystyle \{z\in \mathbb {C} \mid |f(z)|\leq |f(0)|\}}$ ${\displaystyle f}$

For the central statement, develop in , d. H., ${\displaystyle f}$ ${\displaystyle z}$

${\displaystyle f(z+w)=b_{0}+b_{1}w+\dotsb +b_{n}w^{n}}$.

Is , so is a zero. Otherwise you select the smallest with and consider the two inequalities for ${\displaystyle b_{0}=0}$ ${\displaystyle z}$ ${\displaystyle k>0}$ ${\displaystyle b_{k}\neq 0}$ ${\displaystyle s\geq 0}$

${\displaystyle |b_{0}|\geq |b_{k}|\,s^{k}}$and . ${\displaystyle |b_{k}|/2\geq |b_{k+1}|\,s+\dotsb +|b_{n}|\,s^{n-k}}$

Both inequalities are true , and there is a finite, greatest one , so that they are true over the entire interval . For one of this interval, choose a with and so that the relationship holds with a real factor . The triangle inequality now applies to the amount of interest of the function value ${\displaystyle s=0}$ ${\displaystyle {\bar {s}}}$ ${\displaystyle s\in [0,{\bar {s}}]}$ ${\displaystyle s}$ ${\displaystyle w\in \mathbb {C} }$ ${\displaystyle |w|=s}$ ${\displaystyle 1\geq c>0}$ ${\displaystyle b_{k}w^{k}=-c\,b_{0}}$

${\displaystyle {\begin{aligned}|f(z+w)|\leq &|b_{0}+b_{k}w^{k}|+\sum _{j>k}|b_{j}|\,|w|^{j}&=&(1-c)|b_{0}|+\sum _{j>k}|b_{j}|\,s^{j}\\[.3em]=&|b_{0}|-s^{k}\left(|b_{k}|-\sum _{j>k}|b_{j}|\,s^{j-k}\right)&\leq &|f(z)|-{\tfrac {1}{2}}\,|b_{k}|\,s^{k}\end{aligned}}}$ .

Proof with methods of topology

A proof with this method was given by Gauss in 1799. He divided the polynomial function in real and imaginary parts . The zero sets of and are composed of individual one-dimensional arcs that connect a finite number of nodes in the plane. An even number of arcs emanate from each node. In no case can an arc simply end at one point. On every circle with a sufficiently large radius there are zeros of and zeros of , which alternate. Each connected part of the zero point graph of has an even number of intersections on a large circle, which include an odd number of intersections of the zero point graph of . An arc of the graph of must therefore protrude from the connected section of the graph of . This only works if the graphs of and intersect, but the point of intersection is a zero of . ${\displaystyle f(x+{\text{i}}y)=u(x,y)+{\text{i}}v(x,y)}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle 2n}$ ${\displaystyle u}$ ${\displaystyle 2n}$ ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle f(z)}$

Modern versions of this proof use the notion of the number of turns. It is assumed that the polynomial has no complex zeros. Then there can be a closed, continuous curve for each ${\displaystyle f(z)}$ ${\displaystyle s>0}$

${\displaystyle \gamma _{s}\colon [0,2\pi ]\to \mathbb {C} }$, ${\displaystyle \gamma _{s}(t)=\min(1,s)^{-n}\,f(s\,e^{{\text{i}}t})}$

that runs through the (scaled) function values ​​of the polynomial on the circle with radius . Since no function value is zero, a rotation number can be defined. Since the curve changes continuously when the parameter changes, the number of revolutions can only change when the changing curve crosses the zero point. Since, according to the assumption, the function has no zero, such a crossing of the zero point is not possible. Therefore the number of turns must be the same for all . ${\displaystyle s}$ ${\displaystyle s}$ ${\displaystyle f(z)}$ ${\displaystyle s>0}$

For very large values ​​of , the curve of the corresponding curve of the -th power, more precisely the polynomial , becomes more and more similar, the number of revolutions must therefore be constant . For very small values ​​of , the curve becomes more and more similar to the constant curve with value , so the number of revolutions - for all constant - must have the value 0 at the same time. At the same time, this is only possible if the following applies, i.e. the polynomial is constant. For higher order polynomials this argument leads to a contradiction, so it has zeros with enter. ${\displaystyle s}$ ${\displaystyle n}$ ${\displaystyle a_{n}z^{n}}$ ${\displaystyle n}$ ${\displaystyle s}$ ${\displaystyle a_{0}}$ ${\displaystyle s>0}$ ${\displaystyle n=0}$ ${\displaystyle z}$ ${\displaystyle f(z)=0}$

Proof with the intermediate value theorem and algebraic methods

Such a proof was presented by Gauss in 1815. It is used that, according to the intermediate value theorem, every real polynomial of odd degree has at least one zero, and that quadratic equations, even with complex coefficients, can be solved elementarily. The proof takes place as a complete induction on the power of the factor in the degree of the polynomial. ${\displaystyle 2}$

It is initially assumed to be square-free and with real coefficients. The degree has a factorization with odd. The proof takes place as a complete induction on the power of the factor in the degree of the polynomial. If there is a zero after the interim value record. It is now assumed in the induction step that all polynomials with degrees with odd have at least one zero. ${\displaystyle f(z)}$ ${\displaystyle n=k\,2^{m}}$ ${\displaystyle k}$ ${\displaystyle m}$ ${\displaystyle 2}$ ${\displaystyle m=0}$ ${\displaystyle k'\,2^{m-1}}$ ${\displaystyle k'}$

For the sake of simplicity, let us construct an (abstract) decay field of the polynomial in which it has the pairwise different (again abstract) zeros , ${\displaystyle \mathbb {K} \supset \mathbb {R} }$ ${\displaystyle f(z)}$ ${\displaystyle z_{1},\dotsc ,z_{n}}$

${\displaystyle f(z)=(z-z_{1})(z-z_{2})\dotsm (z-z_{n})}$.

In is the set of points , will be considered. Since the abstract zeros are different in pairs, there is only a finite number of straight lines that run through at least two of these points, in particular only a finite number of real slopes of such straight lines for which the difference takes the same value twice. For all other values ​​of is the polynomial ${\displaystyle \mathbb {K} \times \mathbb {K} }$ ${\displaystyle {\tfrac {1}{2}}n(n-1)}$ ${\displaystyle (z_{j}+z_{k},z_{j}\,z_{k})}$ ${\displaystyle j<k}$ ${\displaystyle m}$ ${\displaystyle z_{j}\,z_{k}-m\,(z_{j}+z_{k})}$ ${\displaystyle m}$

${\displaystyle g_{m}(x)=\prod _{j<k}(x-m\,(z_{j}+z_{k})+z_{j}\,z_{k})}$

also square-free and symmetrical in the abstract zeros . The coefficients of can therefore be represented as polynomials in and the coefficients of , i.e. for each real one it is a polynomial with real coefficients and can be determined using resultants from . The degree of is , where is an odd number. According to the induction hypothesis, there is at least one complex zero with . Complex numbers and can be determined from the partial derivatives to and in the zero , so that at least one of the zeros of is a zero of . ${\displaystyle z_{1},\dotsc ,z_{n}}$ ${\displaystyle g_{m}(x)}$ ${\displaystyle m}$ ${\displaystyle f(z)}$ ${\displaystyle g_{m}(x)}$ ${\displaystyle m}$ ${\displaystyle f(z)}$ ${\displaystyle g_{m}(x)}$ ${\displaystyle 2^{m-1}\,k(n-1)}$ ${\displaystyle k(n-1)}$ ${\displaystyle x}$ ${\displaystyle g_{m}(x)=0}$ ${\displaystyle m}$ ${\displaystyle x}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle z^{2}+p\,z+q=0}$ ${\displaystyle f(z)}$

Has also really complex coefficients so has only real coefficients. Every zero of the product is the zero of a factor, thus a zero of itself or as a complex conjugate number . If the now real polynomial is not square-free, then with polynomial arithmetic (e.g. Euclidean algorithm ) a factorization into (non-constant) square-free factors can be found, each of which contains at least one zero. ${\displaystyle f(z)}$ ${\displaystyle {\tilde {f}}(z)={\bar {f}}(z)\,f(z)}$ ${\displaystyle f(z)}$

Proof with methods of function theory

Proof with Liouville's theorem

Because there is a so that applies to everyone with . Because both the absolute value and the absolute value are continuous and the circular disk is compact, according to Weierstrass' theorem there is a place with the minimum absolute value of the function value for all . According to construction there is even a global minimum. If it were positive, the reciprocal function would be holomorphically bounded on and through , i.e. constant according to Liouville's theorem . This would also be constant, which contradicts the prerequisite. It follows that there is a zero (in ). ${\displaystyle \textstyle \lim _{s\to \infty }\inf _{|z|=s}\left|f(z)\right|=\infty }$ ${\displaystyle R>0}$ ${\displaystyle \left|f(0)\right|\leq \left|f(z)\right|}$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle |z|>R}$ ${\displaystyle f}$ ${\displaystyle |f|}$ ${\displaystyle {\overline {U_{R}(0)}}}$ ${\displaystyle z_{0}\in {\overline {U_{R}(0)}}}$ ${\displaystyle C:=\left|f(z_{0})\right|\leq \left|f(z)\right|}$ ${\displaystyle z\in {\overline {U_{R}(0)}}}$ ${\displaystyle C=\left|f(z_{0})\right|}$ ${\displaystyle C}$ ${\displaystyle z\mapsto {\tfrac {1}{f(z)}}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle {\tfrac {1}{C}}}$ ${\displaystyle f(z)}$ ${\displaystyle C\geq 0}$ ${\displaystyle C=\left|f(z_{0})\right|=0}$ ${\displaystyle z_{0}}$

Proof directly using Cauchy's integral theorem

The fundamental theorem of algebra can even be derived directly from Cauchy's integral theorem with the help of elementary estimates , as follows:

The polynomial can be represented in the form , where is another polynomial. ${\displaystyle f}$ ${\displaystyle f(z)=z\cdot g(z)+a_{0}}$ ${\displaystyle g}$

If one now assumes that there is no zero , one can write forever: ${\displaystyle f(z)}$ ${\displaystyle z\in \mathbb {C} \setminus \{0\}}$

${\displaystyle {\tfrac {1}{z}}={\tfrac {f(z)}{z\cdot f(z)}}={\tfrac {g(z)}{f(z)}}+{\tfrac {a_{0}}{z\cdot f(z)}}}$.

Now for each one forms the path integral of the reciprocal function formed on the circular path and obtains: ${\displaystyle R\in \mathbb {N} }$ ${\displaystyle \mathbb {C} \setminus \{0\}}$ ${\displaystyle z\mapsto {\tfrac {1}{z}}}$ ${\displaystyle {\gamma _{R}}(t)=R\cdot \exp(i\cdot t)(t\in [0,2\pi ])}$

${\displaystyle 2\pi i=\int \limits _{\gamma _{R}}{\tfrac {1}{z}}\,\mathrm {d} z=\int \limits _{\gamma _{R}}{\tfrac {g(z)}{f(z)}}\,\mathrm {d} z+a_{0}\cdot \int \limits _{\gamma _{R}}{\tfrac {1}{z\cdot f(z)}}\,\mathrm {d} z}$.

Because of the assumed freedom from zeros in is ${\displaystyle f(z)}$

${\displaystyle z\mapsto {\tfrac {g(z)}{f(z)}}}$

holomorphic , which further results from Cauchy's integral theorem:

${\displaystyle 2\pi i=0+a_{0}\cdot \int \limits _{\gamma _{R}}{\tfrac {1}{z\cdot f(z)}}\,\mathrm {d} z}$

and it:

${\displaystyle 2\pi \leq |a_{0}|\cdot \operatorname {L} ({\gamma _{R}})\cdot \max _{|z|=R}{\tfrac {1}{|z\cdot f(z)|}}=|a_{0}|\cdot 2\pi R\cdot \max _{|z|=R}{\tfrac {1}{|z|\cdot |f(z)|}}=|a_{0}|\cdot 2\pi \cdot \max _{|z|=R}{\tfrac {1}{|f(z)|}}}$  .

This applies to any one . ${\displaystyle R\in \mathbb {R} ^{>0}}$

Now, however, and thus it follows immediately from the last inequality : ${\displaystyle \textstyle \lim _{|z|\to \infty }\ |f(z)|=\infty }$

${\displaystyle 2\pi \leq 0}$,

which is certainly wrong.

The assumed freedom from zeros has thus led to a contradiction and must have a zero. ${\displaystyle f(z)}$ ${\displaystyle f(z)}$

Proof with methods of complex geometry

We understand as a representation of the complex-projective space , i. H. , . The mapping of complex manifolds defined in this way is holomorphic and therefore open (i.e. the image of every open subset is open). Since is compact and continuous, the image is also compact , especially completed in . With that the picture is already whole , because it is coherent . In particular, there is one which is mapped to i.e. H. a zero of . ${\displaystyle f(z)}$ ${\displaystyle \mathbb {P} _{\mathbb {C} }^{1}\to \mathbb {P} _{\mathbb {C} }^{1}}$ ${\displaystyle z\in \mathbb {C} \mapsto f(z)\in \mathbb {C} }$ ${\displaystyle \infty \mapsto \infty }$ ${\displaystyle \mathbb {P} _{\mathbb {C} }^{1}}$ ${\displaystyle f}$ ${\displaystyle f(\mathbb {P} _{\mathbb {C} }^{1})}$ ${\displaystyle \mathbb {P} _{\mathbb {C} }^{1}}$ ${\displaystyle \mathbb {P} _{\mathbb {C} }^{1}}$ ${\displaystyle \mathbb {P} _{\mathbb {C} }^{1}}$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle 0}$ ${\displaystyle f}$

Proof with methods of differential topology

Similar to the proof from the complex geometry above, we understand the sphere as a self-image . So is (real) differentiable and the set of critical points is finite as the set of zeros of the derivative, with which the set of regular values ​​is connected. The cardinality of the archetype of a regular value is also locally constant as a function in ( is injective on neighborhoods of points in ). This shows that is surjective, because regular values ​​are thus always assumed and critical values ​​are assumed by definition. ${\displaystyle f}$ ${\displaystyle \mathbb {S} ^{2}}$ ${\displaystyle f\colon \mathbb {S} ^{2}\to \mathbb {S} ^{2}}$ ${\displaystyle |f^{-1}(y)|}$ ${\displaystyle y\in \mathbb {S} ^{2}}$ ${\displaystyle y}$ ${\displaystyle f}$ ${\displaystyle f^{-1}(y)}$ ${\displaystyle f}$

Generalization of the fundamental theorem

The fundamental theorem of algebra can be further generalized with the help of topological methods using the homotopy theory and the degree of mapping :

Every continuous function    for which a natural number   and further a complex number   exist in such a way that is     satisfied has a zero . ${\displaystyle f\colon \mathbb {C} \to \mathbb {C} }$   ${\displaystyle n>0}$   ${\displaystyle c\neq 0}$ ${\displaystyle \textstyle \lim _{z\to \infty }\ {\tfrac {f(z)}{z^{n}}}=c}$

The fundamental theorem follows from this, by taking   the leading coefficient as a constant for a complex polynomial function     of degree   . ${\displaystyle z\mapsto f(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+\dotsb +a_{1}z+a_{0}}$     ${\displaystyle (z\in \mathbb {C} )}$   ${\displaystyle n>0}$ ${\displaystyle c=a_{n}}$

literature

• Carl Friedrich Gauß : Methodvs nova integralivm valores per approximationem inveniendi. Dieterich, Göttingen 1815, ( Another new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree . (PDF; 190 kB) (English translation of the original)).
• Karl Weierstrass : New proof of the theorem that every whole rational function of a variable can be represented as a product of linear functions of the same variable. In: Meeting reports of the Royal Prussian Academy of Sciences in Berlin. 1891, pp. 1085-1101, bbaw.de
• Saugata Basu, Richard Pollack, Marie-Françoise Roy : Algorithms in Real Algebraic Geometry (= Algorithms and Computation in Mathematics. Vol. 10). 2nd Edition. Springer, Berlin a. a. 2006, ISBN 3-540-33098-4 .
• Eberhard Freitag , Rolf Busam: Function Theory 1 . 3rd, revised and expanded edition. Springer-Verlag, Berlin a. a. 2000, ISBN 3-540-67641-4 .
• Egbert Harzheim : Introduction to combinatorial topology (=  mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X ( MR0533264 ).
• Reinhold Remmert : Fundamentalsatz der Algebra , in D. Ebbinghaus u. a. (Ed.), Numbers, Springer, 1983, p. 78ff

Web links

Wikibooks: Proof of the Fundamental Theorem of Algebra  - Learning and Teaching Materials

Individual evidence

1. ↑ Eberhard Freitag , Rolf Busam: Function theory 1 . 3rd, revised and expanded edition. Springer-Verlag, Berlin a. a. 2000, ISBN 3-540-67641-4 , pp. 84 .
2. ^ John W. Milnor: Topology from the Differentiable Viewpoint . S. 8-9 .
3. ↑ See chap. 5, § 3 ( A homotopy-theoretical proof of the Gaussian fundamental theorem of algebra ) in: Egbert Harzheim: Introduction to combinatorial topology . S. 170-175 .