Let $k$ and $l$ be two multiplicatively independent positive integers $b$ an integer with $b\ge2$. $S$ a finite set of integers. Nishioka proved that for any algebraic number $\alpha$ $0<|\alpha|<1$ the infinite products $\prod_{y=0}^{\infty}(1-{\alpha}^{d^{y}})$ ($d=2,3,\ldots$) are algebraically over $\mathbb{Q}$. As her result, example, transcendence $\prod_{y=0}^{\infty}(1-\frac{1}{{b...

We show that the infinite symmetric product of a connected graded-commutative algebra over ℚ is naturally isomorphic to free on positive degree subspace original algebra. In particular, commutative (in usual sense) graded polynomial Applied topology, we obtain quick proof Dold–Thom theorem in rational homotopy theory for spaces finite type. also products certain simple algebras are free; this a...