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by theorems 1（Euler’s Formula） we have the following

Theorem 3.4. Normal extensions remain normal under lifting. If K ⊃ E ⊃ k and K is normal over k, then K is normal over E. If K₁, K₂ are normal over k and are contained in some field L, then K₁K₂ is normal over k, and so is K₁ ∩ K₂.

我们首先来回忆一下扩张的定义

Proof. For our first assertion, let K be normal over k, let F be any extension of k, and assume K, F are contained in some bigger field. Let σ be an embedding of KF over F (in F^a ). Then σ induces the identity on F, hence on k, and by hypothesis its restriction to K maps K into itself. We get (KF)^σ = K^σF^σ = KF whence KF is normal over F.

Assume that K ⊃ E ⊃ k and that K is normal over k. Let σ be an embedding of K over E. Then σ is also an embedding of K over k, and our assertion follows by definition.

Finally, if K₁, K₂ are normal over k, then for any embedding σ of K₁K₂ over k we have σ(K₁K₂) = σ(K₁)σ(K₂) and our assertion again follows from the hypothesis. The assertion concerning the intersection is true because σ(K₁∩K₂) = σ(K₁) ∩ σ(K₂).

We observe that if K is a finitely generated normal extension of k, say K = k(α₁,…,α_n) and p₁, …, p_n are the respective irreducible polynomials of α₁, …, α_n over k then K is already the splitting field of the finite family p₁, …, p_n. We shall investigate later when K is the splitting field of a single irreducible polynomial.

c is an element of X. C is a subset of X. 𝒞 is a collection of subsets of X. ℂ is a superset whose elements are collections of subsets of X. test ends