# Show that any finite purely inseparable extensions has a $p-$basis.

A set $\{a_1,...,a_n\} \subseteq K$ is said to be a $p-$basis for $K/F$ provided that there is a chain of proper extensions

$F \subset F(a_1) \subset \cdots \subset F(a_n)=K$.

Show that any finite purely inseparable extensions has a $p-$basis.

Now my intution is I can treat the purely inseparable extension as a simple extension then I am done. But Is this intution true then how to prove this? If not then how should I approach to this problem?

• I don't think every purely inseparable extension is simple, e.g., $F$ is the rational functions in variables $u$ and $v$ over the field of $p$ elements, and $K$ is the extension by the $p$-th roots of $u$ and $v$. – Gerry Myerson Oct 20 '15 at 12:45
• Are you sure you have stated the question correctly? $F(a_n)=K$ can't happen, for the $F$ and $K$ in the example I gave. – Gerry Myerson Oct 21 '15 at 5:42
• The extension I gave has degree $p^2$. It's finite. I did find the definition in the book you mention. It's wrong. It's wrong. It's wrong. It's wrong. It's wrong. Just because it got published in a book, that doesn't mean it's right. Books have mistakes. It's wrong. It's wrong. It's wrong. It's wrong. It's wrong. – Gerry Myerson Oct 25 '15 at 0:23
• Possible duplicate of Show that if $\{a_1, ... ,a_n\}$ is a $p$-basis for $K/F$, then $[K: F] = p^n$. – Brahadeesh Sep 11 '18 at 11:08
• This is a known error in the book. The inclusions should read $F \subset F(a_1) \subset \dots \subset F(a_1,\dots,a_{n-1}) \subset F(a_1,\dots,a_n) = K$. See web.nmsu.edu/~pamorand/Errata.pdf – Brahadeesh Sep 11 '18 at 11:10

The question as stated is incorrect without additional hypotheses on the extension $K/F$. For, let $k = \mathbb{F}_p$, and define $F = k(x^p,y^p)$ and $K = k(x,y)$. Then, $K/F$ is a finite purely inseparable extension (in fact $[K:F] = p^2$) which is not simple. So, there is no $p$-basis for $K$ as per the definition given in the question details.
However, this is a known error in the textbook. The inclusions should read $$F \subset F(a_1) \subset F(a_1,a_2) \subset \dots \subset F(a_1,\dots,a_n) = K.$$ See here for a list of errata.
The corrected version of this question is asked and answered here: Show that if $\{a_1, ... ,a_n\}$ is a $p$-basis for $K/F$, then $[K: F] = p^n$.