Eilenberg–Niven theorem

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}xa_{1}x\cdots xa_{n}+\varphi (x)

where x, 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) has at least one solution.^[1]

Generalizations[]

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[]

^ Eilenberg, Samuel; Niven, Ivan (April 1944). "The "fundamental theorem of algebra" for quaternions". Bulletin of the American Mathematical Society. 50 (4): 246–248.
^ 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.
^ 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.

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[1] Eilenberg, Samuel; Niven, Ivan (April 1944). "The "fundamental theorem of algebra" for quaternions". Bulletin of the American Mathematical Society. 50 (4): 246–248.

[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. 27 (2): 1531–1550. doi:10.1007/s00006-016-0748-9. ISSN 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. 17 (2): 245–258. doi:10.1007/s00006-007-0026-y. ISSN 0188-7009.

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