Question

Let F be a field of characteristic 0 , let f(x)∈F[x], and let K/F be a splitting field for f(x) over F. This exercise asks you to prove Proposition 9.34, which states the K is the splitting field of a separable polynomial in F[x] (a) We know from Corollary 7.20 that we can factor f(x) as a product of irreducible polynomials, say f(x)=cg1​(x)^e1​g2​(x)^e2​⋯gr​(x)er​ where g1​(x),…,gr​(x)∈F[x] are distinct monic irreducible polynomials. Prove that gi​(x) and gj​(x) have a common root ⟺i=j. (b) Let g(x)=g1​(x)g2​(x)⋯gr​(x). Prove that g(x) is a separable polynomial. (c) Prove that K is the splitting field of g(x) over F.

Answer 1

(a) If gi(x) and gj(x) share a common root, then i must equal j, as distinct monic irreducible polynomials have distinct roots.

(b) The polynomial g(x) formed by multiplying distinct monic irreducible polynomials is separable.

(c) K is the splitting field of g(x) over F.

Step-by-step explanation:

(a) In the context of monic irreducible polynomials, the common root of gi(x) and gj(x) implies that they are the same polynomial, leading to the conclusion that i must equal j. This is because distinct monic irreducible polynomials have distinct roots, as established by the factorization of f(x) into irreducible components.

(b) The separability of g(x) is a consequence of the distinctness of its irreducible components. When multiplying distinct monic irreducible polynomials, the resulting polynomial g(x) is free from repeated roots, making it separable.

(c) The splitting field K of g(x) over F is established by the distinct roots of the irreducible components of g(x). Each irreducible polynomial contributes roots to K, ensuring that K contains all the roots of g(x) and is therefore its splitting field.