Field theory and polynomials Algebraic closure Theorem

Fundamental theorem of algebra

Builds on Complex numbers Liouville's theorem

Statement

Let $n \ge 1$ be an integer and let $a_0, a_1, \dots, a_n$ be complex numbers with $a_n \neq 0$ . Let $p(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$ . Then there is a complex number $z_0$ with $p(z_0) = 0$ .

Remarks

Proof with Liouville's theorem: if $p$ has no zero, then $1/p$ is entire. Since $|p(z)| \to \infty$ as $|z| \to \infty$ , $1/p$ is bounded, so it is constant, which is false for $n \ge 1$ . By induction on $n$ , it follows that $p(z) = a_n (z - z_1) \cdots (z - z_n)$ for some $z_1, \dots, z_n \in \mathbb{C}$ . So $p$ has exactly $n$ roots, counted with multiplicity. The theorem holds for polynomials with real coefficients too, but the roots can be non-real: $z^2 + 1$ has the roots $i$ and $-i$ .

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Complex numbers object
Real numbers object
Set bounded above definition
Complex exponential definition
Contour integral definition
Power series in the complex plane definition
Real and imaginary parts definition
Supremum definition
Circle integral definition
Disc of convergence definition
Completeness axiom statement
Estimation lemma statement
Existence of square roots of nonnegative reals statement
Square root definition
Modulus of a complex number definition
Complex differentiability definition
Holomorphic function definition
Cauchy's integral formula statement
Entire function definition
Power series are holomorphic in the disc of convergence statement
Holomorphic functions are analytic statement
Holomorphic functions are infinitely differentiable statement
Cauchy's integral formula for derivatives statement
Cauchy's estimates statement
Liouville's theorem statement
Fundamental theorem of algebra statement

Where this leads

3 pages build on this one.

Tag	`0JP0W07Y9A`
URI	`m2://0JP0W07Y9A`
Aliases	`dalembert-gauss-theorem`, `dalemberts-theorem`
Kind	statement, theorem
Writing	W1 stated
MSC	12D10
Status	This copy of the site has no build data. Run `m2 status 0JP0W07Y9A`.
Uses	Complex numbers, Liouville's theorem
Same as	Wikidata Q192760 Wikipedia Fundamental theorem of algebra

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