
Chapter 23
Algebraically Closed Fields
Back to square one.
In Chapter 2 we proved the Fundamental Theorem of Algebra, Theorem 2.4,
using some basic point-set topology and simple estimates. It is also possible to give
an ‘almost’ algebraic proof, in which the only extraneous information required is
that every polynomial of odd degree over R has a real zero. This follows immediately
from the continuity of polynomials over R and the fact that an odd degree polynomial
changes sign somewhere between −∞ and +∞.
We now present this almost-algebraic proof, which applies to a slight general-
isation. The main property of R that we require is that R is an ordered ...