By David B Surowski

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2 (i) Let E ⊇ E1 ⊇ E2 ⊇ F, and assume that [E1 : E2 ] < ∞ and that E2 is closed in E. Then E1 is closed in E. (ii) Let {e} ≤ H1 ≤ H2 ≤ G and assume that [H2 : H1 ] < ∞ and that H1 is closed in G. Then H2 is closed in G. 4 Let E ⊇ F be an algebraic extension with F closed in E. Then every element of ΩE/F is closed in E. The field extension E ⊇ F is called a Galois extension (we sometimes say that E is Galois over F) if F is closed in E. Let E ⊇ F be a field extension with Galois group G, and let K ∈ ΩE/F .

12. Let F ⊆ E1 , E2 ⊆ E be fields. Define E1 E2 ⊆ E to be the smallest field containing both E1 and E2 . E1 E2 is called the composite (or compositum) of the fields E1 and E2 . Prove that if [E : F] < ∞, then [E1 E2 : F] ≤ [E1 : F] · [E2 : F]. 13. Given a complex number α it can be quite difficult to determine whether α is algebraic or transcendental. It was known already in the nineteenth century that π and √ e are transcendental, but the fact that such numbers as eπ and 2 2 are transcendental is more recent, and follows from the following deep theorem of Gelfond and Schneider: Let α and β be algebraic numbers.

2 Let F1 be a field, let f1 (x) ∈ F1 [x], and let K1 be a splitting field over F1 for f1 (x). Let ∼ = ψ : F1 −→ F2 , ˆ 1 (x)) ∈ F2 [x], and let K2 be a splitting field over F2 for f2 (x). let f2 (x) = ψ(f Then there is a commutative diagram K1 ✻ F1 ψ¯ ✲ K2 ✻ ψ ✲ K2 where the vertical maps are inclusions, and where ψ¯ is an isomorphism. Let F be a field and let F ⊆ F[x]. By a splitting field for F we mean a field extension K ⊇ F such that every polynomial in F splits completely in K, and K is minimal in this respect.

### A workbook in higher algebra by David B Surowski

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