Download Workbook in Higher Algebra(en)(194s) by Surowski D. PDF

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Prove that Γ ◦ ι ◦ Γ = Γ, and that ι ◦ Γ ◦ ι = ι. Thus images under either map are always closed. √ 3. Let α = 4 2 ∈ R, and set K = Q(α). Compute the closure of Q in K. 54 CHAPTER 2. FIELD AND GALOIS THEORY 4. 2, let f (x) = x3 + x2 − 2x − 1 ∈ Q[x], and let α ∈ C be a root of f (x). Compute the closure of Q in Q(α). 5. If E ⊇ F is a finite Galois extension, prove that every subgroup of G = Gal(E/F) is closed. 6. Let E ⊇ K ⊇ F with E ⊇ F algebraic. If E is Galois over K and K is Galois over F, must it be true that E is Galois over F?

5 Let F ⊆ K be an algebraic extension of fields where F has characteristic p > 0. If α ∈ K, then α is separable over F if and only if F(α) = F(αp ). 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.

B) 2 + ζ, where ζ = e2πi/3 . 2. Let F ⊆ K be a field extension with [K : F] odd. If α ∈ K, prove that F(α2 ) = F(α). 3. Assume that α = a + bi ∈ C is algebraic over Q, where a is rational and b is real. Prove that mα (x) has even degree. √ √ 4. Let K = Q( 3 2, 2) ⊆ C. Compute [K : Q]. √ 5. Let K = Q( 4 2, i) ⊆ C. Show that (a) K contains all roots of x4 − 2 ∈ Q[x]. (b) Compute [K : Q]. 46 CHAPTER 2. FIELD AND GALOIS THEORY 6. Let F = C(x), where C is the complex number field and x is an indeterminate.