since is a root of in ,
we can apply the division algorithm in , to get:
clearly f splits over F iff does.
since char(F) = 7 ≠ 2, we can use the quadratic formula to obtain the roots:
so x will be in F iff 4 is a square in F. and indeed, in F, both 2 and 5 (that is, -2), are square roots of 4, giving us the two roots .
(here, the "±" is just to indicate that we have (possibly) two square roots, there is no clear reason to call one "positive).
to verify, note .
iii) , which means that