Remark 2

Let $E$ be an algebraic extension of $K$, of infinite degree. By Th. 2 there exists an infinite sequence $(a_i)$, of elements of $E$ such that $a_i \notin K(a_1, a_2, ..., a_{i-1})$; Th. 2 shows further that the degree of $K(a_1, a_2, ..., a_n)$ over $K$ takes arbitrarily large values. In other words, if E is an algebraic extension of K such that the degrees [F : K] of the subextensions F of E of finite degree over K are bounded, then E is an extension of finite degree of K.