Fundamental Theorem of Algebra

The fundamental theorem of algebra is a theorem stating that every nonconstant, single-variable polynomial with complex coefficients has at least one complex root. Equivalently, a polynomial of degree $n\geq1$ has exactly $n$ complex roots when repetitions are counted according to multiplicity. In the language of abstract algebra, it says that the complex numbers form an algebraically closed field. The theorem establishes that complex numbers suffice to solve all polynomial equations in one variable, although it does not supply a formula for their solutions. (jmilne.org)

Statement and factorization

Let

p(z)=a_nz^n+a_{n-1}z^{n-1}+\cdots+a_0, \qquad a_n\neq0,\quad n\geq1,

where all coefficients belong to $\mathbb C$ . The theorem asserts that some $\alpha\in\mathbb C$ satisfies $p(\alpha)=0$ . Once this root exists, the factor theorem gives

p(z)=(z-\alpha)q(z),

where $q$ has degree $n-1$ . Repeating the argument yields

p(z)=a_n\prod_{j=1}^{n}(z-\alpha_j).

Thus the existence statement implies complete factorization into linear factors. The factors are unique up to their order when the leading coefficient is fixed. (jmilne.org)

A root has multiplicity $m$ if $(z-\alpha)^m$ divides $p(z)$ , but $(z-\alpha)^{m+1}$ does not. For example,

z^3-3z+2=(z-1)^2(z+2)

has two distinct roots, but three roots counted with multiplicity: $1,1,-2$ . Nonzero constant polynomials have no roots and are excluded; the zero polynomial vanishes everywhere and has no degree to which the root count applies. (linear.axler.net)

Complex and real coefficients

The theorem concerns the field $\mathbb C$ , not every possible coefficient field. Over the real numbers, $z^2+1$ has no root; over $\mathbb C$ , it factors as $(z-i)(z+i)$ . Complex coefficients are also permitted: they need not be real. (jmilne.org)

For a polynomial with real coefficients, complex conjugation gives

p(\overline z)=\overline{p(z)}.

Consequently, nonreal roots occur in conjugate pairs with equal multiplicity. Combining each pair produces a real quadratic factor:

(z-\alpha)(z-\overline\alpha) =z^2-2\operatorname{Re}(\alpha)z+|\alpha|^2.

Every nonconstant real polynomial therefore factors over $\mathbb R$ into linear factors and irreducible quadratic factors. In particular, every real polynomial of odd degree has a real root. (math.ucla.edu)

Proof methods

Despite its name, the theorem connects algebra with analysis. Standard proofs use properties of the real or complex numbers beyond formal polynomial manipulation. Different approaches highlight different aspects of this connection. (math.ucla.edu)

A short proof from complex analysis uses Liouville’s theorem, which states that a bounded entire function is constant. Suppose $p$ has no root. Then $1/p$ is complex differentiable everywhere. Since the leading term dominates for large $|z|$ , one has $|p(z)|\to\infty$ , so $1/p(z)\to0$ . On any fixed closed disk, $1/p$ is a continuous function and is bounded by the extreme value theorem. It is therefore bounded on the whole plane. Liouville’s theorem makes $1/p$ constant, contradicting the assumption that $p$ is nonconstant. (users.math.msu.edu)

Another proof minimizes $|p(z)|$ . Its growth at infinity ensures that a global minimum is attained. If the minimum occurred at $z_0$ with $p(z_0)\neq0$ , the first nonzero term after the constant in the expansion of $p(z_0+w)$ would permit a small displacement $w$ that decreases the modulus. This contradicts minimality. An algebraic approach instead uses Galois theory, together with the facts that positive real numbers have square roots and odd-degree real polynomials have real roots. (linear.axler.net)

Historical development

Eighteenth-century attempts included work by d’Alembert, Euler, and Lagrange. These arguments helped develop the theorem, but relied on assumptions that were not adequately justified. Carl Friedrich Gauss presented a new proof in his doctoral dissertation of 1799. It treated real-coefficient polynomials through the curves on which the real and imaginary parts vanish. Although influential, this proof contained a significant gap concerning the behavior of those curves. Describing it unconditionally as the first fully rigorous proof obscures that difficulty. (mathshistory.st-andrews.ac.uk)

Gauss subsequently published further proofs in 1816 and 1849. Argand published an argument in 1806 and a further treatment in 1814, contributing to the minimum-modulus approach and the formulation for complex coefficients. The history reflects the gradual clarification of complex numbers, continuity, and existence proofs rather than a single uncontested discovery. (mathshistory.st-andrews.ac.uk)

Consequences and limitations

In linear algebra, an $n\times n$ complex matrix has a degree- $n$ characteristic polynomial. Its roots are the matrix’s eigenvalues, so every such matrix with $n\geq1$ has an eigenvalue. This does not imply diagonalizability: repeated eigenvalues need not provide enough linearly independent eigenvectors. (linear.axler.net)

Existence is distinct from expression by radicals. The Abel–Ruffini theorem rules out a general radical formula for polynomial equations of degree five or higher, without contradicting the existence of their complex roots. The fundamental theorem also does not, by itself, specify a computational procedure for locating those roots. Its assertion is about existence and factorization, not a universal explicit solution formula. (jmilne.org)