The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.152.3. 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 10.152.2. 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. $\square$


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