Eilenberg–Niven theorem
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
[edit]Let
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.[1]
Generalizations
[edit]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]
References
[edit]- ^ Eilenberg, Samuel; Niven, Ivan (April 1944). "The "fundamental theorem of algebra" for quaternions". Bulletin of the American Mathematical Society. 50 (4): 246–248. doi:10.1090/S0002-9904-1944-08125-1.
- ^ Liu, Ming-Sheng; Xiang, Na; Yang, Yan (2017). "On the Zeroes of Clifford Algebra-Valued Polynomials with Paravector Coefficients". Advances in Applied Clifford Algebras. 27 (2): 1531–1550. doi:10.1007/s00006-016-0748-9. ISSN 0188-7009. S2CID 253598676.
- ^ Jou, Yuh-Lin (1950). "The "fundamental theorem of algebra" for Cayley numbers". Acad. Sinica Science Record. 3: 29–33.
- ^ Serôdio, Rogério (2007). "On Octonionic Polynomials". Advances in Applied Clifford Algebras. 17 (2): 245–258. doi:10.1007/s00006-007-0026-y. ISSN 0188-7009. S2CID 123578310.
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