Exercise 12, § 11

Let K be a field of characteristic q, $\Omega$ an algebraically closed extension of K, x an element of $\Omega$ such that $x \notin K$ and M a maximal element of the set of subextensions of $\Omega$ not containing x (V, p. 156, Ex. 9); suppose further that M is perfect so that $M(x)$ is a Galois extension of prime degree p of M (loc. cit.). a) Show that either $p = 2$ and $\Omega = M(i)$ or for each integer r $\geq 1$, there exists a single extension of degree $p^r$ of M; this extension M, is cyclic over M, $\Omega$ is the union of the M, and the M, are the only extensions of finite degree of M. (Note that if a p-group $\Gamma$ of order $p^r$ has only one subgroup of order $p^{r-1}$, it is necessarily cyclic, arguing by induction on r and using I, p. 133, Ex. 10, and I, p. 77, Cor. of Prop. 11). b) Suppose that $\Omega \neq M(i)$ and that $p \neq q$. Show that M contains the p-th roots of unity; we thus have $M(x) = M_1 = M(\alpha)$ with $\alpha^p \in M$. Show that if a is the p-th power of an element of $M_1$, then $M_1 = M(\zeta)$, where $\zeta$ is $p^2$-th root of unity (if $a = \beta^p$ for $\beta \in M_1$, consider the minimal polynomial of $\beta$ in $M[X]$). c) Under the hypotheses of b) show that if a is not a p-th power of an element of $M_1$ then $M_r$ is the splitting field of the polynomial $X^{p^{r-1}} - a$ of $M_1[X]$ (use b)). d) Under the hypotheses of b), show that if a is the p-th power of an element of $M_1$, there exists $y \in M_1$ such that $\gamma \notin M$ and that $M_2$ is the splitting field of $X^p - \gamma$; show that y is not the p-th power of an element of $M_2$ and deduce that $M_r$ is the splitting field of the polynomial $X^{p^{r-1}} - \gamma$ of $M_1[X]$.