The Eilenberg–Niven theorem is a theorem that generalizes the fundamental theorem of algebra to quaternionic polynomials; that is, polynomials with quaternion coefficients and variables. It is due to Samuel Eilenberg and Ivan M. Niven.
Statement Let
P ( x ) = a 0 x a 1 x ⋯ x a n + φ ( x ) , {\displaystyle P(x)=a_{0}xa_{1}x\cdots xa_{n}+\varphi (x),}
where x, a0, a1, ... , an 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: for a counterexample, P(x) = x + ixi + 1 = 0 has no solutions. The Eilenberg–Niven theorem can also be generalized to octonions: all octonionic polynomials with a unique monomial of highest degree have at least one solution, independent of the order of the parentheses (the octonions are a non-associative algebra). Different from quaternions, however, the monic and non-monic octonionic polynomials do not have always the same set of zeros.
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