The Fundamental Theorem of Field Theory
A field
is an extension of a field
if
The set of all expressions
where
is an extension of
The Fundamental Theorem of Field Theory says that a polynomial always has a zero in some extension field. Concisely,
Letbe a field and let
be a nonconstant polynomial in
- so that that coefficients of
are elements of
There is an extension fieldofin whichhas a zero.
Proof
Sinceis a unique factorisation domain,has an irreducible factor,
It is enough to construct an extension fieldofin whichhas a zero. A candidate for
is
This is a field and the mapping
given by
is one to one and preserves operations, so thathas a subfield isomorphic toWe can think ofas containingby thinking as
as the coset
To show that
has a zero in
write
Then
Example: Let
In
we have
