📂Abstract Algebra

Algebraic Numbers and Transcendental Numbers

Definition ¹

Let’s call the field extension of field $F$ as $E$. For a non-constant function $f(x) \in F [ x ]$, if it satisfies $f( \alpha ) = 0$ for $\alpha \in E$, it is called algebraic over $F$, and if not, it is called transcendental. When $F = \mathbb{Q}$, $E = \mathbb{C}$, if $\alpha \in \mathbb{C}$ is algebraic, it is called an algebraic number, and if transcendental, a transcendental number.

Explanation

For example, if there is a polynomial $ f(x) = x^2 - 2 $, there may be no rational root that satisfies $f(x) = 0$, but in the extended field $\mathbb{R}$ from $\mathbb{Q}$, there exists a root $\sqrt{2}$. However, for numbers like $\pi$, they cannot be derived in this manner. Thus, although both $\sqrt{2}$ and $\pi$ are irrational numbers, $\sqrt{2}$ is an algebraic number, and $\pi$ is a transcendental number.

Surprisingly, the concepts of algebraic and transcendental numbers are familiar starting from high school. It is often heard that what distinguishes arts and sciences in high school is the calculus of transcendental functions, which usually comes with explanations about algebraic and transcendental numbers.

At the high school level, it is commonly explained that if it can be a solution to a polynomial equation with integer coefficients, it is an algebraic number; otherwise, it is a transcendental number. This can be neatly summarized as $F = \mathbb{Q}$, $E = \mathbb{C}$ when speaking in the language of abstract algebra.