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Let $L/K$ be a finite separable extension of fields. Then we have the primitive element theorem, i.e., there exists an $x$ in $L$ such that $L=K(x)$.

In particular, the primitive element theorem holds for all finite extensions of a perfect field.

Question 1. Is a field $K$ perfect if and only if the primitive element theorem holds for all finite extensions of $K$?

Question 2. Suppose that $K$ is a field extension of $\mathbf{F}_p$ of transcendence degree 1, i.e., a function field over a finite field. Does the primitive element theorem hold for any finite extension of $K$?

In Question 2, I am actually only interested in the case $K=\mathbf{F}_q(t)$.

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1 Answer 1

up vote 8 down vote accepted

Let $K$ be a field of characteristic $p\neq 0$ and assume that the degree $[K:K^p]$ is finite. Then it is equal to $p^m$ for some $m\in\mathbb{N}$. A result of Becker and MacLane (THE MINIMUM NUMBER OF GENERATORS FOR INSEPARABLE ALGEBRAIC EXTENSIONS, Bull. AMS 46 (2), 1940) says that any finite extension $L/K$ can be generated by at most $\max (1,m)$ elements.

Now $\mathbb{F}_p(t)^p =\mathbb{F}_p(t^p)$ hence $[\mathbb{F}_p(t):\mathbb{F}_p(t)^p]=p$. Consequently every finite extension of $\mathbb{F}_p(t)$ has a primitive element. Since this field is not perfect the answer to question 1 is "no". Note that one can replace $\mathbb{F}_p$ by any perfect field. Also, it easily follows that the answer to question 2 is "yes".

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Great! So given a function field over a perfect field $K$, the primitive element theorem holds for $K$. –  Ali Jan 23 '12 at 9:39
1  
+1: this is a fantastically erudite answer and I strongly advise all users to upvote it. Dear @Hagen: will you excuse my curiosity if I ask you how you discovered Becker and MacLane's result ? –  Georges Elencwajg Jan 24 '12 at 20:46
    
@Georges: Thank you for the laurels. Actually I remembered that Koch did something on generators of purely inseparable extensions. Looking at one of his articles I realized that it was built on the results of Becker and MacLane. So ... –  Hagen Jan 25 '12 at 10:44
    
Thanks for satisfying my curiosity, @Hagen. –  Georges Elencwajg Jan 26 '12 at 11:07

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