Since algebraic extensions have finite subextensions, this theorem is equivalent to there being no (*) nontrivial finite extension of the complex numbers.
One proof of this theorem takes the normal closure of a finite extension and then considers a 2-Sylow subgroup of that extension's Galois group. This theorem follows from continuity and Picard's little theorem. Another proof considers the function 1-over-f, which is entire and bounded, and thus must be constant by Liouville's theorem. Since algebraic extensions have finite subextensions, this theorem is equivalent to there being no (*) nontrivial finite extension of the complex numbers. This theorem implies that irreducible complex polynomials are linear, and thus non-constant complex polynomials are a product of linear factors. For 10 points, name this theorem stating that nth degree polynomials have n complex roots.
One proof of this theorem takes the normal closure of a finite extension and then considers a 2-Sylow subgroup of that extension's Galois group.
fundamental theorem of algebra