Lemma 10.158.2. Let $k$ be a perfect field of characteristic $p > 0$. Let $K/k$ be an extension. Let $a \in K$. Then $\text{d}a = 0$ in $\Omega _{K/k}$ if and only if $a$ is a $p$th power.
Proof. By Lemma 10.131.5 we see that there exists a subfield $k \subset L \subset K$ such that $L/k$ is a finitely generated field extension and such that $\text{d}a$ is zero in $\Omega _{L/k}$. Hence we may assume that $K$ is a finitely generated field extension of $k$.
Choose a transcendence basis $x_1, \ldots , x_ r \in K$ such that $K$ is finite separable over $k(x_1, \ldots , x_ r)$. This is possible by the definitions, see Definitions 10.45.1 and 10.42.1. We remark that the result holds for the purely transcendental subfield $k(x_1, \ldots , x_ r) \subset K$. Namely,
\[ \Omega _{k(x_1, \ldots , x_ r)/k} = \bigoplus \nolimits _{i = 1}^ r k(x_1, \ldots , x_ r) \text{d}x_ i \]
and any rational function all of whose partial derivatives are zero is a $p$th power. Moreover, we also have
\[ \Omega _{K/k} = \bigoplus \nolimits _{i = 1}^ r K\text{d}x_ i \]
since $k(x_1, \ldots , x_ r) \subset K$ is finite separable (computation omitted). Suppose $a \in K$ is an element such that $\text{d}a = 0$ in the module of differentials. By our choice of $x_ i$ we see that the minimal polynomial $P(T) \in k(x_1, \ldots , x_ r)[T]$ of $a$ is separable. Write
\[ P(T) = T^ d + \sum \nolimits _{i = 1}^ d a_ i T^{d - i} \]
and hence
\[ 0 = \text{d}P(a) = \sum \nolimits _{i = 1}^ d a^{d - i}\text{d}a_ i \]
in $\Omega _{K/k}$. By the description of $\Omega _{K/k}$ above and the fact that $P$ was the minimal polynomial of $a$, we see that this implies $\text{d}a_ i = 0$. Hence $a_ i = b_ i^ p$ for each $i$. Therefore by Fields, Lemma 9.28.2 we see that $a$ is a $p$th power. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: