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S 5 Is Not Solvable Unsolvability of Quintics by Radicals Abel’s Theorem Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Introduction logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Introduction 1. If p ( x ) = 0 is solvable by radicals, then the Galois group of p must be solvable. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Introduction 1. If p ( x ) = 0 is solvable by radicals, then the Galois group of p must be solvable. 2. There is an irreducible polynomial with Galois group isomorphic to S 5 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Introduction 1. If p ( x ) = 0 is solvable by radicals, then the Galois group of p must be solvable. 2. There is an irreducible polynomial with Galois group isomorphic to S 5 . 3. But S 5 does not have many normal subgroups, so we have a chance to determine if it is solvable. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. Then H ∩ N is a normal subgroup of N. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. Then H ∩ N is a normal subgroup of N. Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. Then H ∩ N is a normal subgroup of N. Proof. As an intersection of two subgroups logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. Then H ∩ N is a normal subgroup of N. Proof. As an intersection of two subgroups, H ∩ N is a subgroup, too. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. Then H ∩ N is a normal subgroup of N. Proof. As an intersection of two subgroups, H ∩ N is a subgroup, too. Let x ∈ N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. Then H ∩ N is a normal subgroup of N. Proof. As an intersection of two subgroups, H ∩ N is a subgroup, too. Let x ∈ N . Because H is normal in G , we have xHx − 1 = H . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. Then H ∩ N is a normal subgroup of N. Proof. As an intersection of two subgroups, H ∩ N is a subgroup, too. Let x ∈ N . Because H is normal in G , we have xHx − 1 = H . So xhx − 1 ∈ H for all h ∈ H . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. Then H ∩ N is a normal subgroup of N. Proof. As an intersection of two subgroups, H ∩ N is a subgroup, too. Let x ∈ N . Because H is normal in G , we have xHx − 1 = H . So xhx − 1 ∈ H for all h ∈ H . Let y ∈ H ∩ N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. Then H ∩ N is a normal subgroup of N. Proof. As an intersection of two subgroups, H ∩ N is a subgroup, too. Let x ∈ N . Because H is normal in G , we have xHx − 1 = H . So xhx − 1 ∈ H for all h ∈ H . Let y ∈ H ∩ N . By what we just proved, xyx − 1 ∈ H . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. Then H ∩ N is a normal subgroup of N. Proof. As an intersection of two subgroups, H ∩ N is a subgroup, too. Let x ∈ N . Because H is normal in G , we have xHx − 1 = H . So xhx − 1 ∈ H for all h ∈ H . Let y ∈ H ∩ N . By what we just proved, xyx − 1 ∈ H . Moreover, because N is a group, we have xyx − 1 ∈ N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. Then H ∩ N is a normal subgroup of N. Proof. As an intersection of two subgroups, H ∩ N is a subgroup, too. Let x ∈ N . Because H is normal in G , we have xHx − 1 = H . So xhx − 1 ∈ H for all h ∈ H . Let y ∈ H ∩ N . By what we just proved, xyx − 1 ∈ H . Moreover, because N is a group, we have xyx − 1 ∈ N . Thus xyx − 1 ∈ H ∩ N for all y ∈ H ∩ N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. Then H ∩ N is a normal subgroup of N. Proof. As an intersection of two subgroups, H ∩ N is a subgroup, too. Let x ∈ N . Because H is normal in G , we have xHx − 1 = H . So xhx − 1 ∈ H for all h ∈ H . Let y ∈ H ∩ N . By what we just proved, xyx − 1 ∈ H . Moreover, because N is a group, we have xyx − 1 ∈ N . Thus xyx − 1 ∈ H ∩ N for all y ∈ H ∩ N . Because x ∈ N was arbitrary, we conclude that H ∩ N is a normal subgroup of N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Lemma. Let G be a group and let H and N be normal subgroups of G. Then H ∩ N is a normal subgroup of N. Proof. As an intersection of two subgroups, H ∩ N is a subgroup, too. Let x ∈ N . Because H is normal in G , we have xHx − 1 = H . So xhx − 1 ∈ H for all h ∈ H . Let y ∈ H ∩ N . By what we just proved, xyx − 1 ∈ H . Moreover, because N is a group, we have xyx − 1 ∈ N . Thus xyx − 1 ∈ H ∩ N for all y ∈ H ∩ N . Because x ∈ N was arbitrary, we conclude that H ∩ N is a normal subgroup of N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Theorem. For every prime number n ≥ 5 , the group S n is not solvable. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Theorem. For every prime number n ≥ 5 , the group S n is not solvable. Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Theorem. For every prime number n ≥ 5 , the group S n is not solvable. Proof. Let n ≥ 5 be a prime number. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Theorem. For every prime number n ≥ 5 , the group S n is not solvable. Proof. Let n ≥ 5 be a prime number. We first prove that { id } , A n and S n are the only normal subgroups of S n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem

S 5 Is Not Solvable Unsolvability of Quintics by Radicals Theorem. For every prime number n ≥ 5 , the group S n is not solvable. Proof. Let n ≥ 5 be a prime number. We first prove that { id } , A n and S n are the only normal subgroups of S n . Let N ⊳ S n be a normal subgroup of S n with N � = { id } . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Abel’s Theorem