Splitting fields (SF’s)
[Existence and uniqueness, not using algebraic closure]
Theorem 1 + 1/2 [Existence] Assume f (x) ∈ F [x]. Then f (x) has a splitting field
K ⊆ F.
Proof Let n = degf (x), we use induction on n. If f (x) = cx + d and c 6= 0 then F is a
splitting field, it contains the only root (−d)/c of f (x). So let n > 1.
Inductive hypothesis: If g(x) is a polynomial of degree ≤ n − 1 over some field F1 then
it has a splitting field.
Take f (x) of degree n, and let f (x) = f1 (x)f2 (x) where f1 (x) is irreducible. By [Chapter
15, 2.3] there is a field F1 = F (α1 ) where f1 (α1 ) = 0. In F1 [x] we can factorize f (x) =
(x − α1 )g(x), then g(x) has degree < n. By the inductive hypothesis, g(x) has a splitting
Q
field K = F1 (α2 , . . . αn ), where g(x) = c ni=2 (x − αi ) (and 0 6= c ∈ F1 ). Then K :=
F (α1 , . . . , αn ) is a splitting field of f (x) over F .
Theorem 1.2 [Uniqueness ] Let K and E be splitting fields of f (x) over F , then there
is an isomorphism η : K → E extending the identity map of F .
Proof The proof works better if one generalises the statement. We will show by induction
on n
(*) Let σ : F1 → F2 be an isomorphism, and let f (x) ∈ F1 [x]. If K is the SF of f (x) over
F1 and E is the SF of f σ (x) over F2 then there is an isomorphism η : K → E extending
σ with η(α) = β.
If this is proved, the Theorem follows taking F = F1 = F2 and σ = Id.
Let n = degf (x). If n = 1 then K = F1 and E = F2 and there is nothing to do. So assume
n > 1. Take a root α of f (x) in K, and let p(x) ∈ F1 [x] be the minimal polynomial of α.
Then f (x) = p(x) · f˜(x) in F1 [x] (by ch. 15, 3.1). Then f σ (x) = pσ (x) · f˜σ (x) in F2 [x].
Let β ∈ E be a root of pσ (x), then pσ (x) is the minimal polynomial of β over F2 (it is
monic and must be irreducible).
By Lemma 4.2* [Chapter 15] there is an embedding η1 : F1 (α) → E extending σ with
η1 (α) = β. The image of η1 (α) is then F2 (β).
In F1 [x] we have f (x) = (x − α)g(x) and then f σ (x) = (x − β)g η1 (x). Then K is the
splitting field of g(x) over F1 (α) and E is the splitting field of g η1 (x) over F2 (β). The
degree of g(s) is < n, so by the inductive hypothesis there is an isomorphism η : K → E
extending η1 . Then η extends σ.
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