Now, consider the map
^{ . }*v* where *v* is any
(fixed) non-zero column vector such as the transpose of
(1, 0..., 0). When and are an elements of *K* with
^{ . }*v* = ^{ . }*v*, we obtain
( - )^{ . }*v* = 0. But
we have assumed that every non-zero element of *K* is invertible so we
must have
- = 0. In other words this map is * one-to-one* on *K*. Thus *K* is actually isomorphic to a vector
space of rank at most *n* over the rationals. By a suitable change of
basis (and restricting to a submatrix) we may as well assume that the
space *K*^{ . }*v* contains *all* column vectors or equivalently
that *K* has rank *n*. Then *K*^{ . }*w* is the space of all column
vectors for *any* non-zero vector *w*. We will henceforth make
this additional assumption as well.

For any *n*×*n* matrix we have (the Cayley-Hamilton
theorem) that *characteristic* polynomial
ch_{}(*T*) of
degree *n* and
ch_{}() = 0. (In the words of one
mathematician *khudh kaa nahi satisfy karega to kiska satisfy
karega?*(Hindi); if it doesn't satisfy its' own then whose will it
satisfy?). On the other hand, we have the minimal
polynomial
(*T*), which is the polynomial of least degree
with rational coefficients that is satisfied by . If
(*T*) = *P*(*T*)*Q*(*T*), then
*P*()*Q*() = 0. Since,
*P*() and *Q*() are in *K* at least one of them must be
zero thus one of them must be a constant; in other words the minimal
polynomial is *irreducible*. It also follows as before that it
divides the characteristic polynomial. One can show that, under the
hypothesis of the previous paragraph (and the fact the we are working
over rationals; a *perfect* field), there is an element
in *K* whose characteristic polynomial is *irreducible*, i. e.
its characteristic polynomial equals its minimal polynomial. In
particular, the field *K* has a basis over the field
of
rationals of the form 1, , ...,
.

As an example, let us consider the ``construction'' of the field
associated with an irreducible polynomial
*P*(*T*) = *T*^{n} + *a*_{1}*T*^{n - 1} + ... + *a*_{n}. We consider the matrix

=

This has minimal polynomial and characteristic polynomial equal to