Definition:Algebraic Element of Ring Extension
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Definition
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 algebraic over $D$ if and only if:
- $\exists \map f x$ over $D$ such that $\map f x = 0_R$
where $\map f x$ is a non-null polynomial in $x$ over $D$.
Examples
$\sqrt 2$ is Algebraic over $\Z$
- $\sqrt 2$ is an algebraic element over the integers $\Z$.
Also see
- An element of $R$ is said to be transcendental if it is not algebraic.
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 64$. Polynomial rings over an integral domain: Definition $(2)$