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I met the following problem when going through the proof of Lemma 8.10 in GTM 167 Fields and Galois Theory. My main confusion is in dealing with composite fields. Let me state my problem:

Let $K/F$ be a finite and purely inseparable extension and $N/F$ be a finite Galois extension. The book claims that the composite field $NK$ is purely inseparable over $N$.

Here, since $N\cap K$ is both separable and purely inseparable over $F$, we have $N \cap K = F$. Then, by Natural Irrationalities, we know that $NK/K$ is Galois and $Gal(NK/K)=Gal(N/F)$. (I wanted to describe the problem more clearly by drawing a graph, but I failed.)

I try to prove the claim by showing, for any $\alpha\in NK$, there exists some integer $n$ such that $\alpha^{p^n}\in N$ where $p=char(F)>0$. I want to use the fact that "for any $\beta\in K$, there exists some integer $m$ such that $\beta^{p^m}\in F$" since $K/F$ is purely inseparable. But I am still not quite sure how such $\alpha\in NK$ looks like. Could anyone help? Thanks a lot.

Jyrki Lahtonen
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bbbfff
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  • Every finite extension of finite fields is separable, so I removed the tag [tag:finite-fields]. – Jyrki Lahtonen Apr 04 '25 at 14:47
  • I get the feeling that under these assumptions the extensions $K/F$ and $N/F$ are linearly disjoint. Doesn't that imply that an $F$-basis $K$ is also an $N$-basis of $NK$? – Jyrki Lahtonen Apr 04 '25 at 14:52
  • Thanks for clarifying that my question is irrelevant to finite fields. I just checked the definitions for linearly disjoint field extensions, but I cannot figure out what you meant. Would you mind giving more details? Thanks again! – bbbfff Apr 05 '25 at 05:24
  • Another thing, is the approach stated in the last paragraph of my question not applicable to composite fields? It seems raising a general element of a composite field to some power is too complicated. – bbbfff Apr 05 '25 at 05:26
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    This thread may have something useful to your question. I'm not an expert on this, but if a basis $\mathcal{B}{x_1,\ldots,x_m}$ of $K/F$ consists of purely inseparable elements, and is also a basis of $NK/N$, it might help. – Jyrki Lahtonen Apr 05 '25 at 07:53

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A bit of a brute force answer: let $\alpha \in NK$. Write $\alpha = Q(x_1, \ldots, x_n, y_1, \ldots, y_{n'})$ where $x_i \in N$ and $y_i \in K$, and $Q$ is a rational function with coefficients in $F$. Since $K/F$ is purely inseparable, we can take a large enough $M$ so that $y_i^{p^M} \in F$ for all $y_i$. Then we have that $$ \alpha^{p^M} = Q(x_1,\ldots, y_{n'})^{p^M} = Q(x_1^{p^M}, \ldots, y_{n'}^{p^M}) \in N. $$

bbbfff
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regr4444
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