Eilenberg–Niven theorem explained

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} + \varphi(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.^[1]

Generalizations

If permitting multiple monomials with the highest degree, then the theorem does not hold, and P(x) = x + ixi + 1 = 0 is a counterexample with no solutions.

Eilenberg–Niven theorem can also be generalized to octonions: 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).^{[2] [3]} Different from quaternions, however, the monic and non-monic octonionic polynomials do not have always the same set of zeros.^[4]

Notes and References

1. Eilenberg. Samuel. Niven. Ivan. April 1944. The "fundamental theorem of algebra" for quaternions. Bulletin of the American Mathematical Society. 50. 4. 246–248. 10.1090/S0002-9904-1944-08125-1 . free.
2. Liu. Ming-Sheng. Xiang. Na. Yang. Yan. 2017. On the Zeroes of Clifford Algebra-Valued Polynomials with Paravector Coefficients. Advances in Applied Clifford Algebras. en. 27. 2. 1531–1550. 10.1007/s00006-016-0748-9. 253598676 . 0188-7009.
3. Jou. Yuh-Lin. 1950. The "fundamental theorem of algebra" for Cayley numbers. Acad. Sinica Science Record. 3. 29–33.
4. Serôdio. Rogério. 2007. On Octonionic Polynomials. Advances in Applied Clifford Algebras. en. 17. 2. 245–258. 10.1007/s00006-007-0026-y. 123578310 . 0188-7009.