# Splitting field for $x^3+x+1$

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

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GL(4) isn't the same as integers mod 4, so checking 0,1,2,3 is likely not the way. –  coffeemath Nov 22 '12 at 14:45
Yeah I was not meaning $\mathbb{Z}_4$, i just used 0,1,2,3 to represent the elements of the field, which looking back would be some very misleading notation. Thanks –  hmmmm Nov 22 '12 at 15:14

As you said, GF(4) is not a splitting field of $x^3 + x + 1$, because any field over $\mathbb{Z}_2$ containing the roots of $x^3 + x + 1$ must contain at least eight elements.
As mentioned in the comments, you might be confusing GF(4) with $\mathbb{Z}_4$, the ring of integers modulo $4$. The ring $\mathbb{Z}_4$ is not even a field. See here for the addition/multiplication table of GF(4) (also denoted by $\mathbb{F}_4$) and more information.