## 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.

Proof

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

Then

Example: LetInwe have