# Realization and GCD-existence theorem for generalized polynomials

In: Mathematica Slovaca, vol. 60, no. 6
It is shown, that in the ring $F_{\mathbb{Q}}[I]$ 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 \emph{Bezout} only if $I = \emptyset$ or $I$ is a singleton. The arithmetic of the ring $F_{\mathbb{Q}}[I]$ is transferred to the ring $(\mathbf{V},F)[z]$ of generalized polynomials with one indeterminate $z$ over $F$ with exponents from the vector space $\mathbf{V}$ over $\mathbb{Q}$. It is proved that the rings $F_{\mathbb{Q}}[I]$ and $(\mathbf{V},F)[z]$ are isomorphic provided $\dim \mathbf{V} = card I$. It follows, for example, that the rings $(\mathbb{R},F)[z]$ and $(\mathbb{C},F)[{z}]$ of generalized polynomials with one indeterminate with real and complex exponents are isomorphic.