Realization and GCD-existence theorem for generalized polynomials
Ladislav Skula · Mathematica Slovaca · 2010
Abstract It is shown, that in the ring F ℚ[I] of generalized polynomials with several indeterminates from the set I over the field F and with rational exponents, each two elements have a greatest common divisor. On the other hand, this ring is Bezout only if I = O/ or I is a singleton. The arithmetic of the ring F ℚ[I] is transferred to the ring (V, F)[z] of generalized polynomials with one indeterminate z over F with exponents from the vector space V over ℚ. It is proved that the rings F ℚ[I] and (V, F)[z] are isomorphic provided dimV = cardI. It follows, for example, that the rings (ℝ, F)[z] and (ℂ, F)[z] of generalized polynomials with one indeterminate with real and complex exponents are isomorphic.