Eilenberg–Niven theorem

Template:Short description 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

$P(x)=a_{0}x{a}_{1}x\cdots x{a}_n+\phi (x),$

where x, a₀, a₁, ... , 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: 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.

References

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