Exercise 12.3.2

Proof. The extensions $\mathbb{Q}\subset L$ considered by Kronecker are extensions generated by finitely many elements ${\alpha }_1,\cdots ,{\alpha }_n$, so $L=\mathbb{Q}({\alpha }_1,\cdots {\alpha }_n)$.

The result of Steinitz mentioned in the Mathematical Notes to Sections 4.1 says that $L$ can be written in the form

where $m\le n$, ${\beta }_1,\dots ,{\beta }_m$ are algebraically independent over $\mathbb{Q}$, and ${\gamma }_1,\dots ,{\gamma }_l$ are algebraic over $\mathbb{Q}$.

The Theorem of the Primitive Element, applied to the field $K$ with characteristic $0$, gives a primitive element $\gamma \in L$ such that $L=K(\gamma )$. Therefore, $L=\mathbb{Q}({\beta }_1,\dots ,{\beta }_m,\gamma )$, where ${\beta }_1,\dots ,{\beta }_m$ are variables, and $\gamma$ is algebraic over $\mathbb{Q}({\beta }_1,\dots ,{\beta }_m)$.

With the notations used by Kronecker, we obtain (12.31). □