Fundamental Theorem of Algebra

Summary: Euler’s §32 claim that every real polynomial factors into real linear and quadratic factors — an early, non-rigorous statement of the fundamental theorem of algebra over $\mathbb{R}$.

Sources: chapter2

Last updated: 2026-04-23

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Euler’s statement

Whatever number of complex linear factors a polynomial function $Z$ of $z$ may have, they can always be paired in such a way that the product of such pairs is real. (source: chapter2, §32)

Equivalently: every polynomial function of $z$ can be written as a product of real linear and real quadratic factors.

Euler’s argument and its gap

For the special case of four complex linear factors, Euler gives an explicit calculation showing they can be paired into two real quadratic factors (source: chapter2, §31; see complex-conjugate-factors). Beyond that he concedes:

Although the same method of proof is not valid for higher powers, nevertheless, there is no doubt that the same property holds for any number of complex factors. (source: chapter2, §32)

He then admits:

Granted that this has not been proved with complete rigor, still the truth of the statement will be corroborated in what follows, where functions with the form $a+b{z}^n$, $a+b{z}^n+c{z}^{2n}$, $a+b{z}^n+c{z}^{2n}+d{z}^{3n}$, etc. will actually be resolved into such real quadratic factors. (source: chapter2, §32)

So for Euler the result is a working hypothesis supported by explicit computations in special families.

Modern framing

The theorem over $\mathbb{C}$ (every non-constant polynomial has a complex root) is equivalent to Euler’s §32 claim once one observes that complex roots of real polynomials come in conjugate pairs and each conjugate pair $(z-\alpha )(z-\bar{\alpha })$ is a real quadratic.

Rigorous proofs came later — notably Gauss’s dissertation (1799) — long after the Introductio (1748).

How Euler uses the theorem

The §32 decomposition into real linear and quadratic factors is the structural result behind partial-fraction-decomposition: a proper rational function with real coefficients decomposes into a sum of simple fractions of the form $A/(p-qz{)}^k$ and — Euler will show later — simple fractions with real quadratic denominators.

Related pages

• factoring-polynomials
• complex-conjugate-factors
• partial-fraction-decomposition
• chapter-2-on-the-transformation-of-functions