Algebraic and Transcendental Elements in a Field Extension

Algebraic and Transcendental Elements in a Field Extension

 Definition: Let $(F, +, *)$ be a field and let $(K, +, *)$ be a field extension of $F$. An element $u \in K$ is said to be Algebraic over $F$ if there exists a nonzero polynomial $f \in F[x]$ such that $f(u) = 0$. An element $u \in K$ is said to be Transcendental over $F$ if there is no such nonzero polynomial in $F[x]$ satisfying $f(u) = 0$.

In other words, given a field $F$ and a field extension $K$, an element of $u$ of $K$ is algebraic if there exists a nonzero polynomial $f$ with coefficients in $F$ for which $u$ is a root of $f$, and $u$ is transcendental over $F$ otherwise.

For example, consider the field of rational numbers $F = \mathbb{Q}$ and consider the field extension $K = \mathbb{R}$ of real numbers.

Consider the element $u = \sqrt{2} \in \mathbb{R}$. Then the polynomial $f \in \mathbb{Q}[x]$ given by $f(x) = x^2 - 2$ is such that:

(1)
\begin{align} \quad f(u) = f(\sqrt{2}) = (\sqrt{2})^2 - 2 = 0 \end{align}

Therefore $\sqrt{2}$ is algebraic over $\mathbb{Q}$.

In general, it is more difficult to show that an element $u \in K$ is transcendental. For the example above, the element $\pi \in \mathbb{R}$ is transcendental over $\mathbb{Q}$.