The fundamental theorem of algebra #
This file proves that every nonconstant complex polynomial has a root using Liouville's theorem.
As a consequence, the complex numbers are algebraically closed.
We also provide some specific results about the Galois groups of ℚ-polynomials with specific numbers of non-real roots.
We also show that an irreducible real polynomial has degree at most two.
The number of complex roots equals the number of real roots plus the number of roots not fixed by complex conjugation (i.e. with some imaginary component).
An irreducible polynomial of prime degree with two non-real roots has full Galois group.
An irreducible polynomial of prime degree with 1-3 non-real roots has full Galois group.
z is a non-real complex root of a real polynomial,
p is divisible by a quadratic polynomial.