Fundamental Theorem of Algebra. Complex numbers are in a sense perfect while there is little doubt that perfect numbers are complex. Leonhard Euler. Fundamental Theorem of Algebra. The Degree of a Polynomial with one variable is A "root" (or "zero") is where the polynomial is equal to zero. A polynomial of degree 4 will have 4 roots. There are a couple of ways to state the Fundamental Theorem of Algebra. One way So, the theorem is also true for polynomials with real coefficients. Example.


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Fundamental Theorem of Algebra -- from Wolfram MathWorld

The term 'conjugate' had been introduced by Cauchy in Gauss 's criticisms of the Lagrange - Laplace proofs did not seem to find immediate favour in France.

Lagrange fundamental theorem of algebra 2nd Edition of his treatise on equations makes no mention of Gauss 's new proof or criticisms. Even the Edition, edited by Poinsotstill expresses complete satisfaction with the Lagrange - Laplace proofs and no mention of the Gauss criticisms.

In on the 50th anniversary of his first proof!

Fundamental Theorem of Algebra

Gauss produced the first proof that a polynomial equation of fundamental theorem of algebra n with complex coefficients has n complex roots.

The proof is similar to the first proof given by Gauss. However it is adds little since it is straightforward to deduce the result for complex coefficients from the result about polynomials with real coefficients. It is worth fundamental theorem of algebra that despite Gauss 's insistence that one could not assume the existence of roots which were then to be proved reals he did believe, as did everyone at that time, that there existed a whole hierarchy of imaginary quantities of which complex numbers were the simplest.

Gauss called them a shadow of shadows.

It was in searching for such generalisations of the complex numbers that Hamilton discovered the quaternions aroundbut of course the quaternions are not a commutative system.

The first proof that the only commutative algebraic field containing R was given by Weierstrass in his lectures of It was published in Hankel 's fundamental theorem of algebra Theorie der complexen Zahlensysteme.

Fundamental Theorem of Algebra

Of course the proofs described above all become valid once one has the modern result that there is a splitting field for every polynomial. He argues that each branch must be continuous, and therefore the curve on each branch fundamental theorem of algebra pass from one asymptote to another.


Furthermore, the asymptotes occur alternately, so no matter how they are paired, they must cross at some point. Fundamental theorem of algebra showed this by pointing out that, if we are to avoid any crossings, then any two connected red asymptotes must completely enclose an even number of blue asymptotes so they can be connected in pairs without crossing the red connectionand likewise any two connected blue asymptotes must enclose an even fundamental theorem of algebra of red asymptotes, and so on.

But eventually we must arrive at a single enclosed asymptote, so it must intersect the enclosing curve.

Fund theorem of algebra

This completes the proof. A rough illustration of this is shown in the left-hand figure below. Now, as r fundamental theorem of algebra zero, the value of f z goes to c0, and when r is exactly zero, the locus of z values is just a single point at the origin, and the corresponding locus of f z values is the single point at c0.

Assuming c0 is not equal to zero, these are two different points.


It follows that as r fundamental theorem of algebra varied continuously from large values as in the left-hand figure to small values as in the right-hand figurethe blue locus of f z values must be dragged across the origin.

In fact, all n of the loops must cross the origin, and the z values for which this occurs are the n complex roots of the polynomial.


Joseph Shipman showed in that the assumption that odd degree polynomials have roots is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed so "odd" can be replaced by "odd prime" and furthermore this holds for fields of all characteristics.

For axiomatization of algebraically fundamental theorem of algebra fields, this is the best possible, as there are counterexamples if a single prime is excluded.

Fundamental theorem of algebra

Mohsen Aliabadi generalized Shipman's result for any field inproving that the sufficient condition for an arbitrary field of any characteristic to be algebraically closed is having a root for any polynomial of fundamental theorem of algebra degree. Every one would agree it's rather a complex matter to write down a number in excess of digits.

Allowing for a pun, if there are odd perfect numbers they may legitimately be called complex.