So in class we had an exercise that was:
Show that $\mathcal{GL}(4)$ is not the splitting field for $x^3+x+1$
Now in the lecture this was done by noting that if $\alpha$ was a root then $Z_2[\alpha]\subset \mathcal{GL}(4)$ but then $|Z_2[\alpha]|=8$ so this is not the case.
However could we not simply notice that if $\mathcal{GL}(4)$ was a splitting field then we would have $x^3+x+1=(x-a)(x-b)(x-c)$- so it would have $3$ roots in $\mathcal{GL}(4)$ but then we can simply check for the elements $0,1,2,3$ that this is not the case? Or have I misunderstood something badly?
Thanks for any help