(Exercises are taken from Introduction to Abstract Algebra by Christopher J Leininger.)

Exercise 5.2.2 Prove that for any field F and any nonconstant polynomial f € F[z], there exists a field L such that

f factors into linear factors over L. Hints: Write f as a product of irreducible factors, and apply Theorem 5.2.2 to

one of those factors to produce FC K₁. Repeat and induct appropriately.