What is the fundamental theorem of algebra?

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Equivalently, a polynomial of degree n has exactly n complex roots when counted with multiplicity, meaning the field of complex numbers is algebraically closed. Despite its misleading name, the theorem cannot be proven using pure algebra alone and requires transcendental methods.

What we know

• Every non-constant single-variable polynomial with complex coefficients has at least one complex root. [2]
• Every non-zero, single-variable, degree n polynomial with complex coefficients has exactly n complex roots when counted with multiplicity. [2]
• Carl Friedrich Gauss produced the second and third complete proofs of the theorem. [4]
• The theorem cannot be proven using only algebra, requiring transcendental methods such as topology or complex function theory. [2]

Where to go next

• whoWho was Carl Friedrich Gauss?
• a case in pointWhat is a complex conjugate root?
• the bigger pictureWhat is a zero of a function?
• how it workedWhy does the theorem require calculus and topology?
• how we knowHow do polynomial roots relate to graph intercepts?
• an unexpected connectionHow does a math theorem connect to the Earth's magnetic field?Carl Friedrich Gauss is famous for proving the fundamental theorem of algebra while also pioneering the study of geophysics and providing the first absolute measurements of the Earth's magnetic field.

Sources

• [1]fundamental theorem of algebra (Wikidata Q192760) · CC0 1.0
• [2]Fundamental theorem of algebra · CC BY-SA 4.0
• [3]Zero of a function · CC BY-SA 4.0
• [4]Carl Friedrich Gauss · CC BY-SA 4.0
• [5]Complex conjugate root theorem · CC BY-SA 4.0