Galois Theory, 5th Edition

Chapter 23Algebraically Closed Fields

DOI: 10.1201/9781003213949-23

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 $-\infty$ and $+\infty$.

We now present this almost-algebraic proof, which applies to a slight generalisation. The main property of $ℝ$ that we require is that $ℝ$ is an ordered field, with a relation ≤ that satsfies the usual properties. ...