Show that there do not exist matrices over such that and .

If is such a matrix, then the minimal polynomial of divides but not ; so the minimal polynomial of divides . Since has dimension 3, the characteristic polynomial of has degree 3, so the minimal polynomial has degree at most 3. Note that is Eisenstein at 2, and so is irreducible over . (See Proposition 13 on page 309 in D&F, and Example 3 on page 310.) So we have a contradiction- no divisor of over can have degree between 1 and 3. So no such matrix exists.