Definition:Transcendental (Abstract Algebra)

Let $$\left({R, +, \circ}\right)$$ be a commutative ring whose zero is $$0_R$$ and whose unity is $$1_R$$.

Let $$\left({D, +, \circ}\right)$$ be an integral domain such that $$D$$ is a subring of $$R$$.

Let $$x \in R$$.

Then $$x$$ is transcendental over $$D$$ iff $$\forall n \in \mathbb{Z}_+: \sum_{k=0}^n a_k \circ x^k = 0_R \Longrightarrow \forall k: 0 \le k \le n: a_k = 0_R$$.

That is, $$x$$ is transcendental over $$D$$ iff there is no way to express $$0_R$$ as a polynomial in $$x$$ over $$D$$ except trivially.

Algebraic
Then $$x$$ is algebraic over $$D$$ iff it is not transcendental over $$D$$.

Ring of Polynomial Forms
When $$x$$ is transcendental over $$D$$, it is conventional to denote it in uppercase, that is, as $$X$$.

Thus $$D \left[{X}\right]$$ is a ring of polynomials over $$D$$ in an element that is transcendental over $$D$$.

The ring $$D \left[{X}\right]$$ is called the ring of polynomial forms over $$D$$.