Definition:Algebraic Element of Ring Extension

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.

• Definition:Integral Element of Ring Extension

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 64$. Polynomial rings over an integral domain: Definition $(2)$