Any Element is Algebraic Over Field Containing It

Theorem

If \(\alpha \in \mathbb{F}\) for field \(\mathbb{F}\), then \(\alpha\) is algebraic over \(\mathbb{F}\).

This is a trivial result, but is nice to have something to explicitly reference when used.

Proof

Clearly if \(\alpha \in \mathbb{F}\) then \(f(X) = X - \alpha \in \mathbb{F}[X]\), and \(f(\alpha) = \alpha - \alpha = 0\).