Proposition 2

— If an extension E of K is of finite degree n, then it is algebraic and the degree over K of each element of E divides n.

For, given $x \in E$, $[K(x):K]$ is finite and divides n ($V$, p. 10, Cor. 1) and hence $x$ is algebraic over $K$ ($V$, p. 17, Cor. 1).

There exist algebraic extensions of infinite degree, for example the algebraic closure of a finite field (V, p. 24, Remark 4).