General Polynomials over Division Algebras and Left Eigenvalues

Details General Polynomials over Division Algebras and Left Eigenvalues General Polynomials over Division Algebras and Left Eigenvalues

2011-10-10

arxiv.org/abs/1110.2021v2

Mathematics

Rings and Algebras

6 pages, 0 figures

Scientific paper

In this paper, we present an isomorphism between the ring of general polynomials over a division ring of degree $p$ over its center $F$ and the group ring of the free monoid with $p^2$ variables. Using this isomorphism, we define the characteristic polynomial of a matrix over any division algebra, i.e. a general polynomial with one variable over the algebra whose roots are precisely the left eigenvalues. Plus, we show how the left eigenvalues of a $4 \times 4$ matrices over any division algebra can be found by solving a general polynomial equation of degree 6 over that algebra.

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