The fundamental theorem states that every non-constant singlevariable polynomial with complex coefficients has at least one complex
root.
LEARNING OBJECTIVE [ edit ]
State the fundamental theorem of algebra
KEY POINTS [ edit ]
The fundamental theorem of algebra states that every non-constant singlevariable polynomial with complex coefficientshas at least one complex root. This includes
polynomials with real coefficients, since every real number is a complex
numberwith zero imaginary part.
Equivalently (by definition), the fundamental theorem states that the field of complex numbers is
algebraically closed.
The fundamental theorem is also stated as follows: every non-zero, single-variable, degree n
polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The
equivalence of the two statements can be proven through the use of successive polynomial
division.
TERM [ edit ]
multiplicity
the number of values for which a given condition holds
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The fundamental theorem of algebra states that every non-constant single-variable
polynomial with complex coefficients has at least one complex root. This includes
polynomials with real coefficients, since every real number is a complex number with zero
imaginary part. Equivalently (by
definition), the theorem states that the
field of complex numbers is algebraically
closed. The theorem is also stated as
follows: every non-zero, single-variable,
degree n polynomial with complex
coefficients has, counted with multiplicity,
exactly n roots. The equivalence of the two
statements can be proven through the use
of successive polynomial division.
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In spite of its name, there is no purely
algebraic proof of the theorem, since any proof must use the completeness of the reals (or
some other equivalent formulation of completeness), which is not an algebraic concept.
Additionally, it is not fundamental for modern algebra; its name was given at a time when the
study of algebra was mainly concerned with the solutions of polynomial equations with real
or complex coefficients.
Complex-Analytic Proof
Find a closed disk D of radius r centered at the origin such that |p(z)| > |p(0)| whenever |z| ≥
r. The minimum of |p(z)| on D, which must exist since D is compact, is therefore achieved at
some point z0 in the interior of D, but not at any point of its boundary.
The Maximum modulus principle (applied to 1/p(z)) implies then that p(z0) = 0. In other
words, z0 is a zero of p(z).
In mathematics, the maximum modulus principle in complex analysis states that if f is a
holomorphic function, then the modulus cannot exhibit a true local maximum that is
properly within the domain of f. In other words, either f is a constant function, or, for any
point z0 inside the domain of f there exist other points arbitrarily close to z0 at which |f |
takes larger values.
The Maximum Modulus Principle
A plot of the modulus of cos(z) (in red) for z in the unit disk centered at the origin (shown in blue). As
predicted by the fundamental theorem, the maximum of the modulus cannot be inside of the disk (so the
highest value on the red surface is somewhere along its edge).