Transcendental polynomials Consider: "Let $K$ be a field. Then every polynomial $p\in K[X]$ is transcendent over $K$." My question are: 1) How can _polynomial_ be transcendent over something ? I thought that definition applied only for elements of $K$... 2) How can I show the above ? Do I have to find a polynomial "whose coefficients are polynomials" such that $p$ is the root of this greater polynomial ?

Just think about for example the fact that $\mathbb Q(\pi)$ and $\mathbb Q[X]$ are isomorphic. They're both transcendental extensions of $\mathbb Q$.