Definition:Algebraic Element of Ring Extension

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

Let $\left({D, +, \circ}\right)$ be an integral subdomain of $R$.

Let $x \in R$.

Then $x$ is algebraic over $D$ if and only if:

$\exists f \left({x}\right)$ over $D$ such that $f \left({x}\right) = 0$

where $f \left({x}\right)$ is a non-null polynomial in $x$ over $D$.

Also see

  • An element of $R$ is said to be transcendental if it is not algebraic.