Let $L/K$ be a field extension with extension degree $n>1$. We say $L/K$ is simple if $L=K(a)$ for some a in $L$. In this case, $a$ is called a primitive element. My question is now about the dimension of the largest K-subspace of L whose nonzero elements are primitive, in the context of a finite extension of a finite field.
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3$\begingroup$ An element is not simple if and only if it lies in a proper sub-extension. The sub-extensions are parameterised by the divisors of the degree of the extension. Now do inclusion–exclusion. $\endgroup$– LSpiceCommented Sep 9, 2020 at 23:50
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1$\begingroup$ Ah, got it. I was reading the question in the title, rather than the one in the body. I agree that this is less obvious. $\endgroup$– LSpiceCommented Sep 10, 2020 at 16:35
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1$\begingroup$ Every finite extension of a finite field is simple, isn't it? If the degree $n$ is prime, then every element of $L$ that's not in $K$ is a simple element. It should be possible to write $L$ as a direct sum of $K$ and a vector space $V$ of dimension $n-1$ in this case, and all nonzero elements of $V$ will be simple. $\endgroup$– Gerry MyersonCommented Sep 11, 2020 at 3:15
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1$\begingroup$ The standard terminology is that $a$ is then a primitive element. $\endgroup$– YCorCommented Sep 23, 2020 at 21:20
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1$\begingroup$ There still seem to be two different questions being asked— do you want $V\subseteq L$ such that every nonzero $a\in V$ is primitive, or just that every element of some basis for $V$ is primitive? $\endgroup$– Tim CampionCommented Sep 28, 2020 at 13:26
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