Proof. We start by defining the following
map:
Note that this map is clearly a
ring homomorphism. For all
![$ p(x),q(x) \in L[x]$](http://images.planetmath.org:8080/cache/objects/4723/l2h/img13.png "$ p(x),q(x) \in L[x]$")
:
Thus, the
kernel of
is an
ideal of
![$ L[x]$](http://images.planetmath.org:8080/cache/objects/4723/l2h/img17.png "$ L[x]$")
:
Note that the kernel is a
non-zero ideal. This fact relies on the fact that
is a finite extension of fields, and therefore it is an
algebraic extension, so every element of
is a
root of a non-zero polynomial
with
coefficients in
, this is,
.
Moreover, the ring of polynomials ![$ L[x]$](http://images.planetmath.org:8080/cache/objects/4723/l2h/img24.png "$ L[x]$")
is a principal ideal domain (see example of PID). Therefore, the kernel of 
is a principal ideal, generated by some polynomial 
:
Note that the only
units in
![$ L[x]$](http://images.planetmath.org:8080/cache/objects/4723/l2h/img28.png "$ L[x]$")
are the constant polynomials, hence if
is another
generator of
then
Let
be the
leading coefficient of
. We define
, so that the leading coefficient of
is
. Also note that by the previous remark,
is the unique generator of
which is monic.
By construction,
, since 
belongs to the kernel of 
, so it satisfies 
.
Finally, if 
is any polynomial such that 
, then
. Since 
generates this ideal, we know that 
must divide 
(this is property 
).
For the uniqueness, note that any polynomial satisfying 
and 
must be a generator of
, and, as we pointed out, there is a unique monic generator, namely 
.
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
![$ m_{k}(x)\in L[x]$](http://images.planetmath.org:8080/cache/objects/4723/l2h/img3.png "$ m_{k}(x)\in L[x]$"){kind=small}
![$ p(x)\in L[x]$](http://images.planetmath.org:8080/cache/objects/4723/l2h/img6.png "$ p(x)\in L[x]$"){kind=small}
![$\displaystyle \psi\colon L[x] \to K$](http://images.planetmath.org:8080/cache/objects/4723/l2h/img11.png "$\displaystyle \psi\colon L[x] \to K$"){kind=small}
![$\displaystyle \operatorname{Ker}(\psi)=\{ p(x)\in L[x] \mid p(k)=0 \}$](http://images.planetmath.org:8080/cache/objects/4723/l2h/img18.png "$\displaystyle \operatorname{Ker}(\psi)=\{ p(x)\in L[x] \mid p(k)=0 \}$"){kind=small}
