📂Abstract Algebra

Algebraic Numbers and Transcendental Numbers

Definition¹

Let $E$ be an extension field of a field $F$. An $\alpha \in E$ satisfying $f( \alpha ) = 0$ for some non-constant $f(x) \in F [ x ]$ is said to be algebraic over $F$, and transcendental if it is not algebraic. When $F = \mathbb{Q}$ and $E = \mathbb{C}$, an $\alpha \in \mathbb{C}$ is called an algebraic number if it is algebraic, and a transcendental number if it is transcendental.

Explanation

For example, given the polynomial function $ f(x) = x^2 - 2 $, there is no rational solution satisfying $f(x) = 0$, but in $\mathbb{R}$, extended from $\mathbb{Q}$, there exists the solution $\sqrt{2}$. However, a number such as $\pi$ cannot be derived in this way. Hence, while $\sqrt{2}$ and $\pi$ are both irrational numbers, $\sqrt{2}$ is said to be an algebraic number and $\pi$ a transcendental number.

Surprisingly, algebraic numbers and transcendental numbers themselves are concepts that one can encounter familiarly from as early as high school. This is because it is often said that what divides the humanities track from the science track in the typical high school curriculum is the calculus of transcendental functions. There, an explanation of algebraic numbers and transcendental numbers usually comes along with it.

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