Fundamental theorem of algebra

Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.

Peter Roth [de], in his book Arithmetica Philosophica (published in 1608, at Nürnberg, by Johann Lantzenberger),[3] 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 real numbers.

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

Also, Euler pointed out that A first attempt at proving the theorem was made by d'Alembert in 1746, but his proof was incomplete.

In modern terms, Euler, de Foncenex, Lagrange, and Laplace were assuming the existence of a splitting field of the polynomial p(z).

[5] The other one was published by Gauss in 1799 and it was mainly geometric, but it had a topological gap, only filled by Alexander Ostrowski in 1920, as discussed in Smale (1981).

The first textbook containing a proof of the theorem was Cauchy's Cours d'analyse de l'École Royale Polytechnique (1821).

It was Weierstrass who raised for the first time, in the middle of the 19th century, the problem of finding a constructive proof of the fundamental theorem of algebra.

He presented his solution, which amounts in modern terms to a combination of the Durand–Kerner method with the homotopy continuation principle, in 1891.

These statements can be proved from previous factorizations by remarking that, if r is a non-real root of a polynomial with real coefficients, its complex conjugate

is a polynomial of degree two with real coefficients (this is the complex conjugate root theorem).

All proofs below involve some mathematical analysis, or at least the topological concept of continuity of real or complex functions.

[10] Some proofs of the theorem only prove that any non-constant polynomial with real coefficients has some complex root.

are exactly the complex conjugates of the roots of p Many non-algebraic proofs of the theorem use the fact (sometimes called the "growth lemma") that a polynomial function p(z) of degree n whose dominant coefficient is 1 behaves like zn when |z| is large enough.

[11] In other words, for some real-valued a and b, the coefficients of the linear remainder on dividing p(x) by x2 − ax − b simultaneously become zero.

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 of b, all the roots of both Rp(x)(a, b) and Sp(x)(a, b) in the variable a are real-valued and alternating each other (interlacing property).

Topological arguments can be applied on the interlacing property to show that the locus of the roots of Rp(x)(a, b) and Sp(x)(a, b) must intersect for some real-valued a and b < 0.

Another analytic 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.

But the number is also equal to N − n and so N = n. Another complex-analytic proof can be given by combining linear algebra with the Cauchy theorem.

Suppose the minimum of |p(z)| on the whole complex plane is achieved at z0; it was seen at the proof which uses Liouville's theorem that such a number must exist.

As mentioned above, it suffices to check the statement "every non-constant polynomial p(z) with real coefficients has a complex root".

Assuming by way of contradiction that [K:C] > 1, we conclude that the 2-group Gal(K/C) contains a subgroup of index 2, so there exists a subextension M of C of degree 2.

There exists still another way to approach the fundamental theorem of algebra, due to J. M. Almira and A. Romero: by Riemannian geometric arguments.

A Riemannian surface (M, g) is said to be flat if its Gaussian curvature, which we denote by Kg, is identically null.

Then f(z) ≠ 0 for each z in C. Furthermore, We can use this functional equation to prove that g, given by for w in C, and for w ∈ S2\{0}, is a well defined Riemannian metric over the sphere S2 (which we identify with the extended complex plane C ∪ {∞}).

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 an a priori bound: it says that if there are solutions then they lie inside the closed disk of center the origin and radius R∞.

Thus, the modulus of any solution is also bounded by for 1 < p < ∞, and in particular (where we define an to mean 1, which is reasonable since 1 is indeed the n-th coefficient of our polynomial).

Writing the equation as and using the Hölder's inequality we find Now, if p = 1, this is thus In the case 1 < p ≤ ∞, taking into account the summation formula for a geometric progression, we have thus and simplifying, Therefore holds, for all 1 ≤ p ≤ ∞.