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