Theorem 1

— Let E be an extension of K. Then the following conditions are equivalent: a) E possesses a primitive element; b) there exist only a finite number of subextensions of E. These conditions are satisfied when E is a separable extension of finite degree. Suppose first that $E$ possesses a primitive element $x$, and let $f$ be the minimal polynomial of $x$ over K. For each monic polynomial $g \in E[X]$ dividing $f$ in $E[X]$ denote by $E_g$ the subextension of $E$ generated by the coefficients of g. Since the possible polynomials g are finite in number (if $f$ splits in $E[X]$ into a product of r monic irreducible polynomials, the number is bounded by $2^r$), the subextensions $E_g$ are finite in number. To prove b) it is therefore enough to show that every subextension L of $E$ is one of the $E_g$. Now if $L$ is a subextension of E, then we have $L[x] = E$; if g is the minimal polynomial of $x$ over L, we have $[E : L] = \deg(g)$. Besides, g is a divisor off in $L[X]$, hence in $E[X]$; so we have $E_g \subset L$ and $E = E_g[x]$. Since $g(x) = 0$, we have $[E : E_g] \leq \deg(g)$, therefore $[E : E_g] \leq [E : L]$ and so $L = E_g$ as we wished to show.

Next we observe that Condition b) implies that the extension E is of finite degree: by Remark 2 of V, p. 18 it suffices to prove that it is algebraic; now if z is an element of $E$ transcendental over $K$ then the subextensions $K(z^n), n \in \mathbf{N}$ are pairwise distinct.

To show that b) $\Rightarrow$ a) we now distinguish two cases: A) If the field $K$ is finite, the field $E$ is a vector space of finite dimension over K and hence is a finite set. Therefore $^1$ (V, p. 78, Lemma 1) there exists an element $x$ of $E$ generating the multiplicative group of $E$, and we have $E = K[x]$.

B) Suppose now that the field $K$ is infinite. If b) holds, the extension $E$ is of finite degree, so b) can also be expressed by saying that $E$ possesses only a finite number of subalgebras. This being so, the implication $b) \Rightarrow a)$ is a consequence of the following more general proposition (for which the hypothesis that the field $K$ be infinite is indispensable, cf. V, p. 153, Ex. 5 of § 7):