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By Surowski D.

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7 1. Show that H char N G⇒H G. 2. Let H be a subgroup of the group G with G ≤ H. Prove that H G. 3. Let G be a finite group and let P be a 2-Sylow subgroup of G. If M ≤ P is a subgroup of index 2 in P and if τ ∈ G is an involution not conjugate to any element of M , conclude that τ ∈ G (commutator subgroup). [Hint: Look at the action of τ on the set of left cosets of M in G. ] 4. Show that any subgroup of a cyclic group is characteristic. 5. Give an example of a group G and a normal subgroup K such that K isn’t characteristic in G.

If α ∈ K, then α ∈ K. Prove that [K : Q] = ∞. ) 9. Let F be a field, contained as a subring of the integral domain R. If every element of R is algebraic over F, show that R is actually a field. Give an example of a non-integral domain R containing a field F such that every element of R is algebraic over F. Obviously, R cannot be a field. 10. Let F ⊆ K be fields and let f (x), g(x) ∈ F[x] with f (x)|g(x) in K[x]. Prove that f (x)|g(x) in F[x]. 11. Let F ⊆ K be fields and let f (x), g(x) ∈ F[x].

Let ζ = e2πi/7 ∈ C, and let α = ζ + ζ −1 . Show that mα (x) = x3 + x2 − 2x − 1 (as in Exercise 3 above), and that α2 − 2 = ζ 2 + ζ −2 . 5. If ζ = e2πi/11 and α = ζ + ζ −1 , compute mα (x) ∈ Q[x]. 6. Let K ⊆ F be a splitting field for some set F of polynomials in F[x]. Prove that K is algebraic over F. 3. 3 51 Galois Extensions, Galois Groups and the Fundamental Theorem of Galois Theory We begin with the following observation. Let K be a field, let S be a set, and let KS be the set of mappings S → K.

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

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