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On ordered division rings

Ismail M. Idris (2003)

Czechoslovak Mathematical Journal

Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel’s axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under x x a 2 for nonzero a , instead of requiring that positive elements have a positive product. In this work, this type of ordering is studied in the case of a division ring. It is shown that it actually behaves the same as in the commutative...

Polynomials and degrees of maps in real normed algebras

Takis Sakkalis (2020)

Communications in Mathematics

Let 𝒜 be the algebra of quaternions or octonions 𝕆 . In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial f ( t ) 𝒜 [ t ] has a root in 𝒜 . As a consequence, the Jacobian determinant | J ( f ) | is always non-negative in 𝒜 . Moreover, using the idea of the topological degree we show that a regular polynomial g ( t ) over 𝒜 has also a root in 𝒜 . Finally, utilizing multiplication ( * ) in 𝒜 , we prove various results on the topological degree of products...

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