Intermediate field extension

Question

Consider de field $\mathbb F_2$ of two elements. Let $\theta$ be a zero of the irreducible polynomial $t^4+t+1$ and consider the extension $\mathbb F[\theta]$. May question is: Can we find an intermediate field $\mathbb L$ such that $\mathbb F\subsetneqq\mathbb L\subsetneqq\mathbb F[\theta]$?

Answer 1

Hints:

$$\begin{align}&\bullet\;\;\Bbb F[\theta]\cong\Bbb F_{16=2^4}\\{}\\&\bullet\;\;\forall\;\text{prime}\;\;p\;,\;\;\Bbb F_{p^n}\le\Bbb F_{p^m}\iff n\mid m\end{align}$$

Answer 2

Consider $\mathbb L=\{0,1,\theta^2 + \theta ,\theta^2 + \theta+1\}$ and show it is a subfield of $\mathbb F[\theta]$

Why $\theta^2+\theta$ ? Because I know that in the finite field isomorphic to $\mathbb F_{16}$, I will find a subfield isomorphic to $\mathbb F_4$, and $\mathbb F_4=\{0,1,a,a+1\}$ with $a^2=a+1$. So I just have to find a polynomial $P$ in $\theta$ such that $P^2=P+1$

Answer 3

Begin by recalling the fact that for all prime powers $p^n$ the field $\Bbb{F}_{p^n}$ has a (unique) subfield isomorphic to $\Bbb{F}_{p^m}$ if and only if $m\mid n$. This is explained in all texts on finite fields (and has also been explained on this site many times).

In this Q&A pair (that I prepared primarily to introduce people to basic arithmetic and discrete logarithms in small finite fields) it is "shown" that $$ L=\Bbb{F}_2[\theta^5] $$ is the only intermediate field.

This can be seen for example as follows. The polynomial $t^4+t+1$ is known to be primitive. In other words, the element $\theta$ has order fifteen. Therefore $\theta^5$ is primitive cubic root of unity. And the non trivial elements of $\Bbb{F}_4$ are exactly the third primitive roots of unity.

From the linked table (what you call $\theta$ is denoted $\gamma$ there) you see that $\theta^5=\theta^2+\theta$. Thus this is exactly the same generator suggested in Xoff's answer (+1) as well.