Exercise 17, § 2

(a) Let $K$ be an infinite field, $E$ a finite separable algebraic extension of $K$ and $g$ a polynomial of $K[X]$. Show that there exists an element $x \in E$ such that $E = K(x)$ and the minimal polynomial of $x$ over $K$ is relatively prime to $g$ (argue as in Algebra, Chapter V, § 7, no. 7, Proposition 12).

(b) Let $k$ be a field, $A$ a primary $k$-algebra (Chapter IV, § 2, Exercise 32) and $m$ its unique prime ideal. If $A/m$ is a transcendental extension of $k$, there exists in $A$ elements which are not algebraic over $k$. If $A/m$ is algebraic over $k$ and $d$ is an integer at least equal to the separable factor of the degree of $A/m$ over $k$ (and hence any integer if this separable factor is infinite), show that there exists in $A$ elements whose minimal polynomial over $k$ is of degree $\geq d$. If further $A/m$ is not separable over $k$ or if $m \neq 0$, there exists in $A$ an element whose minimal polynomial over $k$ is of degree $> d$. (If $A/m$ is not separable over $k$, use Algebra, Chapter V, § 8, Exercise 5. If $A/m$ is separable and of degree $d$ over $k$ and $x \in A$ is such that $\xi \in A/m$, the class of $x$, is a primitive element of $A/m$, whose minimal polynomial is $f \in k[X]$, observe that $f'(\xi) \neq 0$ and deduce that, if $y \neq 0$ is an element of $m$, then $f(x + y) \neq 0$.)

(c) Let $k$ be an infinite field and $A$ a $k$-algebra the direct composition of $r$ primary algebras $A_i$, $(1 \leq i \leq r)$; let $m_i$ be the maximal ideal of $A_i$, and $d_i$ an integer at least equal to the separable factor of the degree of $A_i/m_i$ over $k$ if $A_i$ is algebraic and this factor is finite, or an arbitrary integer otherwise. Show that there exists in $A$ an element which is not algebraic over $k$ or whose minimal polynomial over $k$ is of degree $\geq \sum_{i=1}^r d_i$; if further one of the $m_i$ is $\neq 0$ or if one of the fields $A_i/m_i$ is not a separable algebraic extension of $k$, there exists in $A$ an element which is not algebraic over $k$ or whose minimal polynomial over $k$ is of degree $> \sum_{i=1}^r d_i$ (use (a) and (b)).