Download A workbook in higher algebra by David B Surowski PDF

By David B Surowski

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The authors actually show that a necessary and sufficient condition for a p-group to be a Cp -group is that all elements of order p are central. In the general case when G is not a p-group, then the authors show that G is a p-group if and only if G has a normal p-Sylow sugroup which is also a p-group. 36 CHAPTER 1. 8 Free Groups; Generators and Relations Let S be a nonempty set and let F be a group. Say that F is free on S if there exists a function φ : S → F such that if G is any group and θ : S → G is any function then there is a unique homomorphism f : F → G such that the diagram φ S ❅ ❅ ❅ θ ❅ ❅ ❘ ✲ F f ✠ G commutes.

Let Ai , i ∈ I be a family of groups. A free product of the groups Ai , i ∈ I is a group P , together with a family of homomorphisms µi : Ai → P , such that if θi : Ai → G is any family of homomorphisms of the groups Ai into a group G, then there exists a unique homomorphism f : P → G making the diagram below commute for each i ∈ I: µi Ai ❅ ❅ ❅ i ❅ ❅ ❘ ✲ P f ✠ G Prove that the free product of the groups Ai , i ∈ I exists and is unique up to isomorphism. (Hint: the uniqueness is just the usual categorical nonsense.

Define the map F : K −→ K, α −→ αq . Then F is easily seen to be an F-automorphism of K, often called the Frobenius automorphism of K. The following is easy to prove. 9 In the notation above, Gal(K/F) is cyclic of order n and is generated by the Frobenius automorphism F . 4 58 CHAPTER 2. FIELD AND GALOIS THEORY 1. Let F ⊆ E be an algebraic Galois extension and let f (x) ∈ F[x] be a separable polynomial. Let K ⊇ E be the splitting field for f (x) over E. Prove that K is Galois over F. 2. Let f (x) = x3 + x2 − 2x − 1 ∈ Q[x], and let K ⊇ Q be a splitting field for f (x) over Q (cf.

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