The Stacks project

Lemma 10.158.3. Let $k$ be a field of characteristic $p > 0$. Let $a_1, \ldots , a_ n \in k$ be elements such that $\text{d}a_1, \ldots , \text{d}a_ n$ are linearly independent in $\Omega _{k/\mathbf{F}_ p}$. Then the field extension $k(a_1^{1/p}, \ldots , a_ n^{1/p})$ has degree $p^ n$ over $k$.

Proof. By induction on $n$. If $n = 1$ the result is Lemma 10.158.2. For the induction step, suppose that $k(a_1^{1/p}, \ldots , a_{n - 1}^{1/p})$ has degree $p^{n - 1}$ over $k$. We have to show that $a_ n$ does not map to a $p$th power in $k(a_1^{1/p}, \ldots , a_{n - 1}^{1/p})$. If it does then we can write

\begin{align*} a_ n & = \left(\sum \nolimits _{I = (i_1, \ldots , i_{n - 1}),\ 0 \leq i_ j \leq p - 1} \lambda _ I a_1^{i_1/p} \ldots a_{n - 1}^{i_{n - 1}/p}\right)^ p \\ & = \sum \nolimits _{I = (i_1, \ldots , i_{n - 1}),\ 0 \leq i_ j \leq p - 1} \lambda _ I^ p a_1^{i_1} \ldots a_{n - 1}^{i_{n - 1}} \end{align*}

Applying $\text{d}$ we see that $\text{d}a_ n$ is linearly dependent on $\text{d}a_ i$, $i < n$. This is a contradiction. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 10.158: Formal smoothness of fields

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Your email address will not be published.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 07DZ. Beware of the difference between the letter 'O' and the digit '0'.