The Eilenberg–Niven theorem is a theorem that generalizes the

fundamental theorem of algebra The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one comp ...

to quaternionic

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s, that is, polynomials with

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...

coefficients and variables. It is due to

Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...

and

Ivan M. Niven Ivan Morton Niven (October 25, 1915 May 9, 1999) was a Canadian-American Number theory, number theorist best remembered for his work on Waring's problem. He worked for many years as a professor at the University of Oregon, and was president of the ...

Statement

Let :

P(x) = a_0 x a_1 x \cdots x a_n + \varphi(x)

where ''x'', ''a''_''0'', ''a''_''1'', ... , ''a''_''n'' are non-zero quaternions and ''φ''(''x'') is a finite sum of monomials similar to the first term but with degree less than ''n''. Then ''P''(''x'') = 0 has at least one solution.

Generalizations

If permitting multiple monomials with the highest degree, then the theorem does not hold, and ''P''(''x'') = ''x'' + i''x''i + 1 = 0 is a counterexample with no solutions. Eilenberg–Niven theorem can also be generalized to

octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...

s: all octonionic polynomials with a unique monomial of higher degree have at least one solution, independent of the order of the parenthesis (the octonions are a

non-associative algebra A non-associative algebra (or distributive algebra) is an algebra over a field where the binary operation, binary multiplication operation is not assumed to be associative operation, associative. That is, an algebraic structure ''A'' is a non-ass ...

). Different from quaternions, however, the monic and non-monic octonionic polynomials do not have always the same set of zeros.

References

Theorems about polynomials {{algebra-stub