6
$\begingroup$

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.

$\endgroup$
11
  • 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. $\endgroup$
    – LSpice
    Sep 9, 2020 at 23:50
  • 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. $\endgroup$
    – LSpice
    Sep 10, 2020 at 16:35
  • 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. $\endgroup$ Sep 11, 2020 at 3:15
  • 1
    $\begingroup$ The standard terminology is that $a$ is then a primitive element. $\endgroup$
    – YCor
    Sep 23, 2020 at 21:20
  • 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 Campion
    Sep 28, 2020 at 13:26

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.