The Stacks project

Lemma 15.46.4. Let $K$ be a field of characteristic $p$. Let $\{ K_\alpha \} _{\alpha \in A}$ be a collection of subfields of $K$ with the following properties

  1. $K^ p \subset K_\alpha $ for all $\alpha \in A$,

  2. $K^ p = \bigcap _{\alpha \in A} K_\alpha $,

  3. for $\alpha , \alpha ' \in A$ there exists an $\alpha '' \in A$ such that $K_{\alpha ''} \subset K_\alpha \cap K_{\alpha '}$.


  1. the intersection of the kernels of the maps $\Omega _{K/\mathbf{F}_ p} \to \Omega _{K/K_\alpha }$ is zero,

  2. for any finite extension $L/K$ we have $L^ p = \bigcap _{\alpha \in A} L^ pK_\alpha $.

Proof. Proof of (1). Choose a $p$-basis $\{ x_ i\} $ for $K$ over $\mathbf{F}_ p$. Suppose that $\eta = \sum _{i \in I'} y_ i \text{d}x_ i$ maps to zero in $\Omega _{K/K_\alpha }$ for every $\alpha \in A$. Here the index set $I'$ is finite. By Lemma 15.46.2 this means that for every $\alpha $ there exists a relation

\[ \sum \nolimits _ E a_{E, \alpha } x^ E,\quad a_{E, \alpha } \in K_\alpha \]

where $E$ runs over multi-indices $E = (e_ i)_{i \in I'}$ with $0 \leq e_ i < p$. On the other hand, Lemma 15.46.2 guarantees there is no such relation $\sum a_ E x^ E = 0$ with $a_ E \in K^ p$. This is a contradiction by Lemma 15.46.3.

Proof of (2). Suppose that we have a tower $L/M/K$ of finite extensions of fields. Set $M_\alpha = M^ p K_\alpha $ and $L_\alpha = L^ p K_\alpha = L^ p M_\alpha $. Then we can first prove that $M^ p = \bigcap _{\alpha \in A} M_\alpha $, and after that prove that $L^ p = \bigcap _{\alpha \in A} L_\alpha $. Hence it suffices to prove (2) for primitive field extensions having no nontrivial subfields. First, assume that $L = K(\theta )$ is separable over $K$. Then $L$ is generated by $\theta ^ p$ over $K$, hence we may assume that $\theta \in L^ p$. In this case we see that

\[ L^ p = K^ p \oplus K^ p\theta \oplus \ldots K^ p\theta ^{d - 1} \quad \text{and}\quad L^ pK_\alpha = K_\alpha \oplus K_\alpha \theta \oplus \ldots K_\alpha \theta ^{d - 1} \]

where $d = [L : K]$. Thus the conclusion is clear in this case. The other case is where $L = K(\theta )$ with $\theta ^ p = t \in K$, $t \not\in K^ p$. In this case we have

\[ L^ p = K^ p \oplus K^ pt \oplus \ldots K^ pt^{p - 1} \quad \text{and}\quad L^ pK_\alpha = K_\alpha \oplus K_\alpha t \oplus \ldots K_\alpha t^{p - 1} \]

Again the result is clear. $\square$

Comments (0)

There are also:

  • 5 comment(s) on Section 15.46: Field extensions, revisited

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07P4. Beware of the difference between the letter 'O' and the digit '0'.