The Fundamental Theorem of Field Theory
A fieldis an extension of a fieldif
The set of all expressionswhereis 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 letbe a nonconstant polynomial in- so that that coefficients ofare elements ofThere is an extension fieldofin whichhas a zero.
Sinceis a unique factorisation domain,has an irreducible factor,It is enough to construct an extension fieldofin whichhas a zero. A candidate for isThis is a field and the mappinggiven byis one to one and preserves operations, so thathas a subfield isomorphic toWe can think ofas containingby thinking asas the coset
To show thathas a zero inwrite
Example: LetInwe have