Fundamental theorem of algebra

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The fundamental theorem of algebra states that every polynomial of degree greater than zero has a root. The domain of the variable is the set of complex numbers, which is an extension of the real numbers.

Although this statement appears to be a weak statement at first, it implies that every polynomial of degree n of a variable with degree greater than zero with complex coefficients has, counting multiplicities, exactly n complex roots. The equivalence of these two statements is made by successive polynomial division by linear factors.

There are many proofs of this important proposition, which require quite a bit of mathematical knowledge to formalize.

History

Pedro Rothe (Petrus Roth), in his book Arithmetica Philosophica (published in 1608), he wrote that a polynomial degree equation $n$ (with real coefficients) can. having to $n$ solutions. Albert Girard, in his book L'invention nouvelle en l'Algebre (published in 1629), he claimed that a grade equation $n$ It has $n$ solutions, but does not mention that such solutions should be real numbers. Moreover, he adds that his assertion is valid "except that the equation is incomplete", which means that none of the coefficients of polynomial is equal to zero. However, when he explains in detail what he is referring to, it becomes clear that the author thinks that the assertion is always true; in particular, he shows that the equation

{displaystyle x^{4}=4,x-3}

despite being incomplete, it has the following four solutions (root 1 has multiplicity 2):

{displaystyle 1,quad 1,quad -1+i,{sqrt {2}}quad mathrm {y} quad -1-i,{sqrt {2}}.}

Leibniz in 1702 and later Nikolaus Bernoulli conjectured otherwise.

As will be mentioned again later, it follows from the fundamental theorem of algebra that all polynomial with real coefficients and grade greater than zero can be written as a product of polynomials with real coefficients of which their grades are 1 or 2. Anyway, in 1702 Leibniz said no polynomial type ${displaystyle x^{4}+a^{4}}$ (with a real and different from 0) can be written in such a way. Then Nikolaus Bernoulli made the same statement concerning polynomial ${displaystyle x^{4}-4x^{3}+2x^{2}+4x+4}$ , received a letter from Euler in 1742 saying that his polynomial was equal to:

{displaystyle (x^{2}-(2+alpha)x+1+{sqrt {7}}+alpha)(x^{2}-(2-alpha)x+1+{sqrt {7}}-alpha)}

with α equal to the square root of 4 + 2√7. He also mentioned that:

{displaystyle x^{4}+a^{4}=(x^{2}+a{sqrt {2}}x+a^{2})(x^{2}-a{sqrt {2}}x+a^{2}).,}

The first attempt to prove the theorem was made by d'Alembert in 1746. His proof was flawed in that it implicitly assumed to be true a theorem (now known as Puiseux's theorem) that would not be true. demonstrated until a century later. Among others Euler (1749), de Foncenex (1759), Lagrange (1772) and Laplace (1795) tried to prove this theorem.

In the late 18th century, two new proofs were presented, one by James Wood and one by Gauss (1799), but both were equally incorrect. Finally, in 1806 Argand published a correct proof for the theorem, stating the fundamental theorem of algebra for polynomials with complex coefficients. Gauss produced another pair of proofs in 1816 and 1849, the latter being another version of his original proof.

The first textbook containing the proof of this theorem was written by Cauchy. This is Course d'anlyse de l'École Royale Polytechnique (1821). The proof is due to Argand, however, credit is not given in the text.

None of the tests mentioned above are constructive. It is Weierstrass who for the first time, in the middle of the 19th century, mentions the problem of finding a constructive proof of the fundamental theorem of algebra. In 1891 he published a demonstration of this type. In 1940 Hellmuth Knesser achieved another proof of this style, which would later be simplified by his son Marin Knesser's in 1981.

Statement and equivalences

The theorem is commonly stated as follows:

All polynomial in a grade variable n ≥ 1 with real or complex coefficients has at least one root (real or complex).

The statement is also widely known: A polynomial in one variable, not constant and with complex coefficients, has as many roots as its degree indicates, counting the roots with their multiplicities. In other words, given a complex polynomial P(z) of degree n ≥ 1, the equation P(z) = 0 has exactly n complex solutions, counting multiplicities.

Other equivalent forms of the theorem are:

The body of the complexes is closed for algebraic operations.
All polynomial degree complex n ≥ 1 can be expressed as a product of n Linear polynomies, i.e.

{displaystyle P(z)=sum _{k=0}^{n}a_{n-k},z^{n-k}=a_{n},prod _{k=1}^{n}(z-z_{k}).}

Demo

Sea $P$ a degree polynomial $n$ . $P$ It's a whole function. For every positive constant $m$ , there is a positive real number $r$ such as

m,quad {mbox{si}}quad |z|>r.}" xmlns="http://www.w3.org/1998/Math/MathML">日本語P(z)日本語▪m,Yeah.日本語z日本語▪r.{displaystyle ΔP(z)healthm,quad {mbox{si}}}quad Δzēr. !m,quad {mbox{si}}quad |z|>r.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/055af9e10495adf08331f5cb79bf4756da531f60" style="vertical-align: -0.838ex; width:25.493ex; height:2.843ex;"/>

Yeah. $P$ has no roots, the function ${displaystyle f=1/P}$ , it is an entire function with the property that for any real number $epsilon$ greater than zero, there is a positive number $r$ such as

<math alttext="{displaystyle |f(z)|r.}" xmlns="http://www.w3.org/1998/Math/MathML">日本語f(z)日本語.ε ε ,Yeah.日本語z日本語▪r.{displaystyle ёf(z) impliesepsilonquad {mbox{si}quad Δz. !<img alt="{displaystyle |f(z)|r.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233dddf4e28981743b662e1c1aac2f2467a51a5d" style="vertical-align: -0.838ex; width:23.93ex; height:2.843ex;"/>

We concluded that the function $f$ It's tied up. But Liouville's theorem says yes $f$ It's a whole and tied function, then, $f$ is constant and this is a contradiction.

So $f$ It's not whole and therefore $P$ has at least one root. $P$ can be written therefore as the product

{displaystyle P(z)=(z-alpha _{1})Q(z),,}

where ${displaystyle alpha _{1}}$ It's a root of $P$ and $Q$ It's a grade polynomial. $n-1$ . For the previous argument, the polynomial $q$ in turn has at least one root and can be factored back.

Repeating this process $n-1$ Sometimes we conclude that the polynomial $P$ can be written as the product

{displaystyle P(z)=k,(z-alpha _{1})(z-alpha _{2})cdots (z-alpha _{n})}

where ${displaystyle alpha _{1}}$ ... ${displaystyle alpha _{n}}$ are the roots of $P$ (not necessarily different) and $k$ It's a constant.

Corollaries

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 theorems concerning algebraically closed fields apply to the field of complex numbers. Some consequences of the theorem are shown here, about the field of real numbers or about the relations between the field of real numbers and the field of complex numbers:

The body of complex numbers is the algebraic closure of the body of real numbers.

All myonic polynomial in a variable $x$ with rational coefficients is the product of a shaped binomial ${displaystyle x+m}$ with $m$ rational, and a trinomial of form ${displaystyle x^{2}+bx+c}$ with $b$ and $c$ rational and $<math alttext="{displaystyle b^{2}-4cb2− − 4c.0{displaystyle b^{2}-4c }<img alt="{displaystyle b^{2}-4c$ (which is the same as saying that trinomial ${displaystyle x^{2}+bx+c}$ is not resoluble in the set of real numbers).

All rational function in a variable $x$ , with real coefficients, can be written as the sum of a polynomial function with rational functions of the form ${displaystyle a/(x-b)^{n}}$ (where) $n$ is a natural number, and $a$ and $b$ are real numbers), and rational functions of form ${displaystyle (ax+b)/(x^{2}+cx+d)^{n}}$ (where) $n$ is a natural number, and $a$ , $b$ , $c$ and $d$ are real numbers such that $<math alttext="{displaystyle c^{2}-4dc2− − 4d.0{displaystyle c^{2}-4d }<img alt="{displaystyle c^{2}-4d$ ). A corollary of this is that any rational function in a real variable and coefficients has an elementary primitive.

Any algebraic extension of the body of the real isomorph to the body of the real or to the body of the complexes.

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