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 pth 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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