Definition:Transcendental (Abstract Algebra)
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Definition
Field Extension
A field extension $E / F$ is said to be transcendental if and only if:
- $\exists \alpha \in E: \alpha$ is transcendental over $F$
That is, a field extension is transcendental if and only if it contains at least one transcendental element.
Transcendental over Integral Domain
Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$.
Let $\struct {D, +, \circ}$ be an integral subdomain of $R$.
Let $x \in R$.
Then $x$ is transcendental over $D$ if and only if:
- $\ds \forall n \in \Z_{\ge 0}: \sum_{k \mathop = 0}^n a_k \circ x^k = 0_R \implies \forall k: 0 \le k \le n: a_k = 0_R$
That is, $x$ is transcendental over $D$ if and only if the only way to express $0_R$ as a polynomial in $x$ over $D$ is by the null polynomial.
Also see
Historical Note
The term transcendental, in the sense of meaning non-algebraic, was introduced by Gottfried Wilhelm von Leibniz.