nLab transcendental extension

Definition

A field extension $K \subset L$ is a transcendental (field) extension of $K$ if there is $\alpha\in L$ for which every polynomial function with coefficients in $K$ is equal to the zero polynomial function if it has $\alpha$ among its roots when interpreted in $L$ (in other words, $\alpha$ is transcendental over $K$).

A field extension $K \subset L$ is a purely transcendental extension if there is an algebraically independent subset $S \subseteq L$ over $K$ such that $K(S)$ is isomorphic to $L$.

Related concepts

• algebraic extension

• transcendental element

• algebraically independent subset

References

• Wikipedia, Transcendental extension