Let $L=\mathbb F_3[X]/(X^3+X^2+2)$ be a field (which has order $3^3=27$). Is there a finite field extension $M\supseteq L$ of degree $n\geq 2$?
Is the following argument correct?
There exists such an extension. Let $f=X^5-1=(X-1)(X^4+X^3+X^2+X+1)\in L[X]$. Then, $f$ only has root $1$ (because $5$ does not divide $|L^\times|=26$); thus, the second factor must have an irreducible, monic factor $g\in L[X]$ of degree $\geq 2$. Now, take $M:=L/(g)$. Then, $[M:L]=\deg(g)\geq 2$.
If it is correct, I was wondering if there a more fundamental or "straight forward" way to prove this. The argument above requries some "guess work" of finding such an irreducible polynomial of degree $\geq 2$; am I missing an easy point here?