Algebraic element

Definition. Let $L/K$ be a field extension and let $\alpha\in L$. The element $\alpha$ is algebraic over $K$ if there exists a nonzero polynomial $f(x)\in K[x]$ such that $f(\alpha)=0$.
If no such nonzero polynomial exists, $\alpha$ is transcendental over K .

Among all polynomials in $K[x]$ vanishing at $\alpha$, there is a unique monic one of minimal degree: the minimal polynomial $m_\alpha(x)$, which is irreducible in $K[x]$.

See also. simple extension , algebraic extension .

Examples.

1. $\sqrt2$ is algebraic over $\mathbb{Q}$ since it satisfies $x^2-2=0$; its minimal polynomial is $x^2-2$.
2. $i$ is algebraic over $\mathbb{R}$ since $i^2+1=0$; $\mathbb{C}=\mathbb{R}(i)$.
3. A primitive n-th root of unity $\zeta_n$ is algebraic over $\mathbb{Q}$ since $\zeta_n^n-1=0$ (more precisely it satisfies the cyclotomic polynomial $\Phi_n(x)$).