Could $X^3 + X + 1$ have a root in $E?$

Question

Consider the irreducible polynomial $g = X^4 + X + 1$ over $F_2$ and let $E$ be the extension of $F_2 =$ {0, 1} with a root $α$ of $g.$

Could $X^3 + X + 1$ have a root in $E?$

When working through this sheet Ive come across this question in which im not sure how to answer so any help will be appreciated

Answer 1

The answer is ‘No’.

If it had a root, say $\beta$, the field $\mathbf F_2(\beta)$ would be a subfield of $\mathbf F_2(\alpha)$. This is impossible, since it would imply $$4=[\mathbf F_2(\alpha)\colon\mathbf F_2]=[\mathbf F_2(\alpha)\colon\mathbf F_2(\beta)]\cdot[\mathbf F_2(\beta)\colon\mathbf F_2]=[\mathbf F_2(\alpha)\colon\mathbf F_2(\beta)]\cdot3.$$

There's a general theorem which asserts that, given a prime $p$ and natural numbers $r, s\ge1$, $$\mathbf F_{p^r}\subset\mathbf F_{p^s}\iff r\mid s.$$