Proof of the fundamental theorem of algebra by Liouville's theorem

Builds on Complex numbers Power series are holomorphic in the disc of convergence Holomorphic function Complex differentiability Entire function Liouville's theorem Triangle inequality Continuous real function on a compact set attains its bounds Heine–Borel theorem

Proof

We argue by contradiction.

Let $n \ge 1$ , let $a_0, \dots, a_n$ be complex numbers with $a_n \ne 0$ , and let $p(z) = \sum_{k=0}^{n} a_k z^k$ .
Assume that $p(z) \ne 0$ for each $z \in \C$ .
The polynomial $p$ is a power series with finitely many nonzero coefficients, so its radius of convergence is $\infty$ . By power series are holomorphic, $p$ is holomorphic on $\C$ .
Then $p$ is continuous: $p(z + h) - p(z) = h \cdot \frac{p(z + h) - p(z)}{h} \to 0 \cdot p'(z) = 0$ as $h \to 0$ .
Let $g = 1/p$ , defined on $\C$ by step 2. For $z \in \C$ and small $h \ne 0$ , $\frac{g(z + h) - g(z)}{h} = - \frac{p(z + h) - p(z)}{h} \cdot \frac{1}{p(z + h) \, p(z)}.$
By steps 3 and 4, the right side of step 5 tends to $-p'(z)/p(z)^2$ . So $g$ is complex differentiable at each point, and $g$ is entire.
Let $S = \sum_{k=0}^{n-1} \abs{a_k}$ and $R_0 = \max(1, 2S/\abs{a_n})$ . Let $\abs{z} \ge R_0$ .
By the triangle inequality, $\abs{p(z)} \ge \abs{a_n} \abs{z}^n - \sum_{k=0}^{n-1} \abs{a_k} \abs{z}^k$ .
Since $\abs{z} \ge 1$ , we have $\abs{z}^k \le \abs{z}^{n-1}$ for $k \le n - 1$ . So $\abs{p(z)} \ge \abs{z}^{n-1} (\abs{a_n} \abs{z} - S)$ .
Since $\abs{z} \ge 2S/\abs{a_n}$ , we have $S \le \abs{a_n} \abs{z} / 2$ . So $\abs{p(z)} \ge \abs{a_n} \abs{z}^n / 2 \ge \abs{a_n}/2$ .
The closed disc $K = \set{z : \abs{z} \le R_0}$ is closed and bounded, so it is compact by the Heine–Borel theorem.
The function $\abs{p}$ is continuous by step 4. It attains its minimum $m$ on $K$ , and $m > 0$ by step 2.
By steps 10 and 12, $\abs{g(z)} \le \max(1/m, 2/\abs{a_n})$ for each $z \in \C$ . So $g$ is bounded.
By Liouville's theorem, $g$ is constant. Hence $p = 1/g$ is constant, and $\abs{p}$ is bounded on $\C$ .
But step 10 gives $\abs{p(z)} \ge \abs{a_n} \abs{z}^n / 2$ for $\abs{z} \ge R_0$ . Since $n \ge 1$ , this tends to $\infty$ as $\abs{z} \to \infty$ .
Steps 14 and 15 contradict each other. So assumption 2 is false, and there is $z_0 \in \C$ with $p(z_0) = 0$ .

Idea

If $p$ had no zero, then $1/p$ would be an entire function. It is small far away, because $\abs{p}$ grows like $\abs{z}^n$ , and it is bounded on a large disc by compactness. Liouville's theorem then makes $1/p$ constant, which a nonconstant polynomial cannot be.

Reading path

To understand this page from the ground up, read these 448 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the derivative of a map into R^n by components, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the diameter characterisation of Cauchy sequences, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the epsilon-delta form of continuity, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Cauchy–Schwarz inequality, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule in several variables, Proof of the chain rule with an auxiliary difference quotient, Proof of the reverse triangle inequality, Proof of the derivative as a little-o approximation, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of continuity via closed sets, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that holomorphic functions are infinitely differentiable, Proof that Lipschitz functions are uniformly continuous, Proof that the integral of a continuous function is an antiderivative, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that linear maps on finite-dimensional normed spaces are continuous, Proof of the limit of a product of sequences, Proof of Cauchy's integral formula by a primitive of the difference quotient, Proof of Cauchy's estimates, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the Cauchy–Riemann equations, Proof of the existence of square roots of nonnegative reals, Proof of the chain rule for a composition with a curve, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof that a continuous image of a compact set is compact, Proof that compact subsets of a Hausdorff space are closed, Proof that the composition of continuous maps is continuous, Proof of epsilon-delta continuity at a point in metric spaces, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof of Cauchy's integral formula for derivatives, Proof of the description of the open sets of a metric space, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that a zero derivative on a connected open set implies constant, Proof of the first fundamental theorem of calculus, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof that the modulus is multiplicative, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof that every spanning set contains a basis, by Zorn's lemma, Proof of the estimation lemma, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that closed sets contain their limits, Proof of the linearity of the integral, Proof of integration by substitution, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof of the second fundamental theorem of calculus, Proof that the complex numbers form a field, Proof of the intermediate value theorem by the supremum, Proof that differentiable functions are continuous, Proof of the sequence characterisation of uniform continuity, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the fundamental theorem of contour integration, Proof of Goursat's theorem by repeated subdivision, Proof of the Archimedean property in an Archimedean ordered field, Proof that a continuous real function on a compact set attains its bounds, Proof of the derivative of a map into a product, Proof that a composition of continuous functions is continuous, Proof of Cantor's intersection theorem, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that holomorphic functions are analytic, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that absolutely convergent series in a Banach space converge, Proof that open balls are open, Proof that power series are holomorphic in the disc of convergence, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that Lipschitz maps are uniformly continuous, Proof that continuous partial derivatives imply differentiability, Proof that a set is open if and only if its complement is closed, Proof of the fundamental theorem of calculus, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that sums and products of continuous functions are continuous, Proof of Liouville's theorem by Cauchy's estimates, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the mean value inequality, Proof of the constant multiple rule.

Basis concept
Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Coordinate space object
Coordinate vector concept
Definition by recursion concept
Element definition
Euclidean metric example
Euclidean space object
Extended real number line object
Function concept
Function composition concept
Group concept
Imaginary unit object
Inner product concept
Inner product space concept
Integers object
Limit point of a set definition
Linear isomorphism concept
Linear map concept
Lipschitz continuity definition
Local maximum definition
Mathematical induction concept
Natural numbers object
Norm concept
Norm induced by an inner product concept
Ordered field definition
Partition of an interval definition
Peano axioms concept
Proof of Zorn's lemma proof
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Vector space definition
p-metric on R^n example
Complex exponential definition
Contour integral definition
Power series in the complex plane definition
Real and imaginary parts definition
Set definition
Fréchet derivative definition
Total derivative concept
Function of several variables concept
Vector-valued function concept
Ring definition
Ring of integers statement
Limit of a function definition
Local minimum definition
Successor function definition
Archimedean ordered field definition
Mesh of a partition definition
Refinement of a partition definition
Riemann sum definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Inner product definition
Linear combination definition
Linear map definition
Linear subspace definition
Norm definition
p-norm on R^n example
Circle integral definition
Complex conjugate definition
Disc of convergence definition
Empty set object
Family of sets definition
Ordered pair definition
Differentiable function of several variables concept
Partial derivative concept
Limit of a function of several variables concept
Parametrised curve concept
Commutative ring definition
Unit of a ring definition
Intersection definition
Union definition
Continuity at a point definition
Derivative definition
One-sided limit definition
Addition of natural numbers definition
Riemann integrable function definition
Biconditional definition
Truth table concept
Modus ponens concept
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
Inner product space definition
Linear independence definition
Composition of linear maps definition
Linear isomorphism definition
Linear span definition
Completeness axiom statement
Normed vector space definition
Integral of 1/z around the unit circle example
Cartesian product definition
Proof of the equality of complex numbers proof
Gradient concept
Jacobian matrix concept
Partial derivative definition
Differentiable function of several variables definition
Field definition
Continuous function definition
Union of a family of sets definition
Subset definition
Antiderivative definition
Differentiable function definition
One-sided derivative definition
Proof of the commutativity of addition by induction proof
Archimedean property statement
Riemann integral definition
Order on the natural numbers definition
Mathematical proof concept
Propositional logic concept
Proof of the derivative as a little-o approximation proof
The rationals form an ordered field statement
Uniqueness quantifier definition
Upper and lower Darboux sums definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Sequential continuity definition
Norm induced by an inner product definition
Partial sum definition
Spanning set definition
Equality of complex numbers statement
Binary relation definition
Universal set definition
Proof of the derivative of a map into R^n by components proof
Topological space definition
Proof that a composition of continuous functions is continuous proof
Order on the reals definition
Derivative as a little-o approximation statement
Derivative function definition
Definite integral definition
Commutativity of addition statement
Tautology definition
Proof of strong induction from ordinary induction proof
Upper and lower Darboux integrals definition
Sum of a finite geometric series statement
Convergent series definition
Finite-dimensional vector space definition
Basis definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Sequential compactness definition
Interval definition
Complement definition
Higher derivative definition
Proof that the complex numbers form a field proof
Subspace topology definition
Function definition
Logical equivalence definition
Coordinate vector definition
Strong induction statement
Sum of a series definition
Absolute convergence definition
Preorder definition
Proof of the epsilon-delta form of the limit of a function proof
Continuously differentiable function definition
Choice function definition
Codomain definition
Function composition definition
Identity function definition
Injective function definition
Proof of the epsilon-delta form of continuity proof
Image of a set definition
Preimage of a set definition
The complex numbers form a field statement
Metric space definition
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Left inverse definition
Right inverse definition
Axiom of choice statement
Epsilon-delta definition of a limit statement
Surjective function definition
Epsilon-delta definition of continuity statement
Closed ball definition
Convergence of a sequence in a metric space definition
Lipschitz map definition
Open ball definition
Triangle inequality for a metric statement
Double negation law statement
Diameter of a set definition
Well-ordering principle statement
Maximal element definition
Total order definition
Upper bound definition
Bijective function definition
Uniform continuity definition
Composition of continuous functions is continuous statement
Metric topology definition
Bounded set definition
Proof by contradiction concept
Proof of the diameter characterisation of Cauchy sequences proof
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Uniform continuity on a set definition
Open set definition
Proof that a function is bijective iff it has a two-sided inverse proof
Epsilon characterisation of Cauchy sequences statement
Proof that Lipschitz functions are uniformly continuous proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Bijective iff it has a two-sided inverse statement
Hausdorff space definition
Closed set definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Continuous map of topological spaces definition
Open cover definition
Equivalent metrics definition
Zorn's lemma statement
Proof of the description of the open sets of a metric space proof
Complete metric space definition
Lipschitz functions are uniformly continuous statement
The reals form an ordered field statement
Connected space definition
Proof that a set is open if and only if its complement is closed proof
Open sets characterised by neighbourhoods statement
Compact space definition
Proof that the composition of continuous maps is continuous proof
Equivalent norms definition
Open sets of a metric space statement
Proof that the coordinate map is a linear isomorphism proof
Absolute value definition
Proof that every spanning set contains a basis, by Zorn's lemma proof
Proof of the approximation property of the supremum proof
Proof of the existence of the infimum by negation proof
Proof of the existence of square roots of nonnegative reals proof
Complement of an open set is closed statement
Connected set definition
Composition of continuous maps is continuous statement
Continuous linear map definition
Continuity at a point of a topological space definition
Compact set definition
Banach space definition
Proof that closed sets contain their limits proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that open balls are open proof
Every spanning set contains a basis statement
Isomorphism with the coordinate space statement
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Existence of the infimum statement
Existence of square roots of nonnegative reals statement
Proof of continuity via closed sets proof
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Closed sets contain their limits statement
Triangle inequality statement
Absolute value is multiplicative statement
Open balls are open statement
Square root definition
Proof of the intermediate value theorem by the supremum proof
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Continuity via closed sets statement
Modulus of a complex number definition
Compactness via open covers statement
Proof of the limit of a sum of sequences proof
Proof of the reverse triangle inequality proof
Proof of the uniqueness of limits proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Intermediate value theorem statement
Proof of continuity via open sets in metric spaces proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof that metric spaces are Hausdorff proof
Proof of the squeeze theorem for sequences proof
Proof that limits preserve non-strict inequalities proof
Proof that differentiable functions are continuous proof
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
The reals are Archimedean statement
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the monotone convergence theorem for sequences proof
Complex differentiability definition
Proof that compact subsets of a Hausdorff space are closed proof
Proof of Cantor's intersection theorem proof
Reverse triangle inequality statement
Proof that a complex number times its conjugate is the square of its modulus proof
Proof that a continuous image of a compact set is compact proof
Epsilon-delta continuity in metric spaces statement
Proof that compact subsets of a metric space are bounded proof
Lebesgue number lemma statement
Metric spaces are Hausdorff statement
Proof that compactness and sequential compactness agree in metric spaces proof
Proof of the sequence characterisation of uniform continuity proof
Differentiable functions are continuous statement
Proof of the triangle inequality for norms proof
Linearity of the integral statement
Limit of a sum of sequences statement
Uniqueness of limits statement
Proof of the monotonicity of the integral proof
Limits preserve non-strict inequalities statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
Subsequences of a convergent sequence converge to the same limit statement
Squeeze theorem statement
Convergent sequences are Cauchy statement
Monotone convergence theorem for sequences statement
Holomorphic function definition
Compact subsets of a Hausdorff space are closed statement
A complex number times its conjugate statement
Cantor's intersection theorem statement
Proof of the Cauchy–Riemann equations proof
Continuous image of a compact set is compact statement
Continuity via open sets statement
Compact subsets of a metric space are bounded statement
Compactness and sequential compactness agree in metric spaces statement
Proof of the limit of a product of sequences proof
Epsilon-delta uniform continuity statement
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
There is a natural number n with 1/n below any positive epsilon statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Triangle inequality for norms statement
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Entire function definition
Cauchy–Riemann equations statement
Proof that the modulus is multiplicative proof
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Proof that Lipschitz maps are uniformly continuous proof
Proof that absolutely convergent series in a Banach space converge proof
Proof of the sequential characterization of limits proof
Proof of the Cauchy–Schwarz inequality proof
Proof that continuity is equivalent to sequential continuity proof
Geometric series example
Proof of Riemann's criterion for integrability proof
Limit of a product of sequences statement
A series of nonnegative terms converges iff its partial sums are bounded statement
Decreasing sequence bounded below converges statement
Modulus is multiplicative statement
Absolutely convergent series in a Banach space converge statement
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Riemann's criterion for integrability statement
Sequential characterization of limits statement
Proof of the ratio test proof
Proof of the comparison test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Cauchy–Schwarz inequality statement
Bounded linear maps are continuous statement
Comparison test statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the linearity of differentiation proof
Proof that sums and products of continuous functions are continuous proof
Proof of the additivity of the integral over intervals proof
Bolzano–Weierstrass theorem statement
Proof of the derivative of a map into a product proof
Product of continuous functions is continuous statement
Additivity of the integral over intervals statement
Chain rule statement
Ratio test statement
Proof of the Heine–Borel theorem by sequential compactness proof
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Operator norm definition
Linearity of differentiation statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Derivative of a map into a product statement
Proof that power series are holomorphic in the disc of convergence proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Heine–Borel theorem statement
Boundedness theorem statement
Proof of the constant multiple rule proof
Proof of the first fundamental theorem of calculus proof
Power series are holomorphic in the disc of convergence statement
Derivative of a map into R^n by components statement
Constant multiple rule statement
Proof that a continuous real function on a compact set attains its bounds proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Extreme value theorem statement
Continuous real function on a compact set attains its bounds statement
Proof of Fermat's theorem on stationary points proof
Proof that continuous functions are Riemann integrable proof
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Equivalence of norms in finite dimension statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of the estimation lemma proof
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Proof of the mean value theorem for integrals proof
Estimation lemma statement
Linear maps on finite-dimensional spaces are continuous statement
Mean value theorem for integrals statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem statement
First fundamental theorem of calculus statement
Proof of the chain rule in several variables proof
Proof that continuous partial derivatives imply differentiability proof
Chain rule in several variables statement
Proof that the integral of a continuous function is an antiderivative proof
Proof of the second fundamental theorem of calculus proof
The integral of a continuous function is an antiderivative statement
Continuous partial derivatives imply differentiability statement
Proof of the chain rule for a composition with a curve proof
Second fundamental theorem of calculus statement
Proof of the fundamental theorem of calculus proof
Chain rule for a composition with a curve statement
Proof of integration by substitution proof
Integration by substitution statement
Fundamental theorem of calculus statement
Proof of the mean value inequality proof
Proof of the fundamental theorem of contour integration proof
Mean value inequality statement
Fundamental theorem of contour integration statement
Proof that a zero derivative on a connected open set implies constant proof
Proof of Goursat's theorem by repeated subdivision proof
Zero derivative on a connected open set implies constant statement
Goursat's theorem statement
Proof of Cauchy's integral formula by a primitive of the difference quotient proof
Cauchy's integral formula statement
Proof that holomorphic functions are analytic proof
Holomorphic functions are analytic statement
Proof that holomorphic functions are infinitely differentiable proof
Holomorphic functions are infinitely differentiable statement
Proof of Cauchy's integral formula for derivatives proof
Cauchy's integral formula for derivatives statement
Proof of Cauchy's estimates proof
Cauchy's estimates statement
Proof of Liouville's theorem by Cauchy's estimates proof
Liouville's theorem statement
Proof of the fundamental theorem of algebra by Liouville's theorem proof

Where this leads

93 pages build on this one.

Tag	`5FBS58NMAK`
URI	`m2://5FBS58NMAK`
Kind	proof
Writing	W0 stub
MSC	12D10
Proves	Fundamental theorem of algebra
Uses	Complex numbers, Power series are holomorphic in the disc of convergence, Holomorphic function, Complex differentiability, Entire function, Liouville's theorem, Triangle inequality, Continuous real function on a compact set attains its bounds, Heine–Borel theorem
Changed	4 November 2026

Proof of the fundamental theorem of algebra by Liouville's theorem

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