Example. $\mathbb{C}, \mathbb{Q}$ and $F(t)$ [1402]

  • $\sqrt{ 2 }, \sqrt[3]{ 2 }, i \in \mathbb{C}$ are algebraic over $\mathbb{Q}$.
  • $e, \pi \in \mathbb{C}$ are transcendental over $\mathbb{Q}$. (nontrivial to prove)
  • Let $F$ be a field. $t \in F(t)$ is transcendental over $F$.