The Fundamental Theorem of Algebra (Algebra 3: Lecture 28 Video 1)

Lecture 28: In the first part of this lecture we gave a proof of the Fundamental Theorem of Algebra. In the second part of the lecture we gave a sketch of the proof that if F is a field of characteristic 0, f(x) in F[x] can be solved by radicals if and only if Gal(f) is a solvable group.

Reading: The proof of the Fundamental Theorem of Algebra we gave comes from pages 615-617 at the end of Section 14.6. This section contains a second proof where the details of the end of the argument are a little different. I encourage you to read that one too. In the second part of the lecture we gave a summary of the main results of the first part of Section 14.7, up to page 629. We did not give full proofs. In particular, we did not say anything about the proofs of Proposition 37 and Lemma 38. We stated the result that follows from the discussion of Cardano's Formulas (pages 630-632) but did not say anything about the proofs.