A workbook in higher algebra by David B Surowski

By David B Surowski

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11. ) Let F be a field and let x be indeterminate over F. Set E = F(x), a simple transcendental extension of F. (i) Let α ∈ E; thus α = f (x)/g(x), where f (x), g(x) ∈ F[x], and where f (x) and g(x) have no common factors. Write n n ai xi , g(x) = f (x) = i=0 bi xi i=0 54 CHAPTER 2. FIELD AND GALOIS THEORY where an = 0, or bn = 0. Therefore, n = max {deg f (x), deg g(x)}. Note that (an − αbn )xn + (an−1 − αbn−1 )xn−1 + · · · + (a0 − αb0 ) = 0. If we set n (ai − αbi )X i ∈ F(α)[X], F (X) = i=0 then x is a root of F (X).

Argue that P = x, y . (b) Show that (yxy −1 )p = xp . (c) Show that yxy −1 ∈ x, Φ(P ) ; conclude that x, yxy −1 is a proper subgroup of P . Thus, by (b), x and yxy −1 commute. 7. THE COMMUTATOR SUBGROUP 35 (d) Conclude from (c) that if z = [x, y] = xyx−1 y −1 , then z p = [xp , y] = [y p , y] = e. Thus, by hypothesis, z commutes with x and y. (e) Show that [y −1 , x] = [x, y]. ) (f) Show that (xy −1 )p = e. 7 24. Here’s an interesting simplicity criterion. Let G be a group acting primitively on the set X, and let H be the stabilizer of some element of X.

We say that f (x) is separable if f (x) has no repeated roots in a splitting field. In general, a polynomial (not necessarily irreducible) is called separable if each is its irreducible factors is separable. Next, if F ⊆ K is a field extension, and if α ∈ K, we say that α is separable over F if α is algebraic over F, and if mα,F (x) is a separable polynomial. Finally we say that the extension F ⊆ K is a separable extension K is algebraic over F and if every element of K is separable over F. 1 Let F ⊆ K be an algebraic extension of fields.

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