[Extension Field Minimal Polynomial][2005-10-23] iamet 发表在 ∑〖数学〗
Given a field and an extension field, if is an algebraic element over , the minimal polynomial of over is the unique monic irreducible polynomial such that . It is the generator of the ideal
of . Any irreducible monic polynomial of has some root in some extension field, so that it is the minimal polynomial of . This arises from the following construction. The quotient ring is a field, since is a maximal ideal, moreover contains . Then is the minimal polynomial of , the residue class of in .
, which is also the simple extension field obtained by adding to . Hence, in this case, and the extension field coincides with the extension ring.
In general, if is any other algebraic element of any extension field of with the same minimal polynomial , it remains true that , and this field is isomorphic to .
and an extension field 
, if 
is an algebraic element over 
, the minimal polynomial of 
over 
is the unique monic irreducible polynomial 
such that 
. It is the generator of the ideal
.
of 
has some root 
in some extension field 
, so that it is the minimal polynomial of 
. This arises from the following construction. The quotient ring 
is a field, since 
is a maximal ideal, moreover 
contains 
. Then 
is the minimal polynomial of 
, the residue class of 
in 
. 
, which is also the simple extension field obtained by adding 
to 
. Hence, in this case,
and the extension field coincides with the extension ring. 
is any other algebraic element of any extension field of 
with the same minimal polynomial 
, it remains true that 
, and this field is isomorphic to 
.