Download e-book for iPad: A workbook in higher algebra by David B Surowski By David B Surowski

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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. If η = log α log β 46 CHAPTER 2. FIELD AND GALOIS THEORY is irrational, then η is transcendental. (See E. Hille, American Mathematical√ Monthly, vol.

If α1 , α2 , · · · , αk are in E, then K := F(α1 , α2 , · · · , αk ) is a finite extension of F, and so H := Gal(E/K) is a subgroup of G of finite index. One easily sees that {O(α1 , α2 , · · · , αk ; σ)} = σH. From this it follows that the basic open sets in the Krull topology on G are precisely the cosets of subgroups of finite index in G. 1 Let σ ∈ G and let µσ : G → G be left multiplication by σ. Then µσ is continuous. 2 Let E ⊇ F be an algebraic Galois extension with Galois group G, and let H ≤ G.

6 Let F ⊆ K be an algebraic extension, where F is a field of characteristic p > 0. Let α ∈ K be an inseparable element over F. The following are equivalent: (i) α is purely inseparable over F. e (ii) The minimal polynomial has the form mα (x) = xp − a ∈ F[x], for some positive integer e and for some a ∈ F. , α. Let F be a field of characteristic p > 0. We may define the p-th power map (·)p : F → F, α → αp . Clearly (·)p is a monomorphism of F into itself. We say that the field F is perfect if one of the following holds: (i) F has characteristic 0, or (ii) F has characteristic p > 0 and (·)p : F → F is an automorphism of F.