Consequences of Solvability by Radicals Bernd Schr oder logo1 - PowerPoint PPT Presentation

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . � � Let σ ∈ G F ( w ) / F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z � � Let σ ∈ G F ( w ) / F logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F is unique modulo n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . � � Let σ , µ ∈ G F ( w ) / F logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . � � Let σ , µ ∈ G F ( w ) / F logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F µ ◦ σ ( w ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F � wr k � µ ◦ σ ( w ) = µ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F � wr k � � r k � = µ ( w ) r k µ ◦ σ ( w ) = µ = µ ( w ) µ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F = µ ( w ) r k = wr j r k � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F = µ ( w ) r k = wr j r k = wr j + k � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F = µ ( w ) r k = wr j r k = wr j + k � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ and hence Φ ( µ ◦ σ ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F = µ ( w ) r k = wr j r k = wr j + k � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ and hence Φ ( µ ◦ σ ) = [ j + k ] n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F = µ ( w ) r k = wr j r k = wr j + k � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ and hence Φ ( µ ◦ σ ) = [ j + k ] n = [ j ] n +[ k ] n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F = µ ( w ) r k = wr j r k = wr j + k � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ and hence Φ ( µ ◦ σ ) = [ j + k ] n = [ j ] n +[ k ] n = Φ ( µ )+ Φ ( σ ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F = µ ( w ) r k = wr j r k = wr j + k � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ and hence Φ ( µ ◦ σ ) = [ j + k ] n = [ j ] n +[ k ] n = Φ ( µ )+ Φ ( σ ) . Moreover, Φ is injective logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F = µ ( w ) r k = wr j r k = wr j + k � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ and hence Φ ( µ ◦ σ ) = [ j + k ] n = [ j ] n +[ k ] n = Φ ( µ )+ Φ ( σ ) . Moreover, Φ is injective, because Φ ( σ ) = [ 0 ] n implies σ ( w ) = w logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F = µ ( w ) r k = wr j r k = wr j + k � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ and hence Φ ( µ ◦ σ ) = [ j + k ] n = [ j ] n +[ k ] n = Φ ( µ )+ Φ ( σ ) . Moreover, Φ is injective, because Φ ( σ ) = [ 0 ] n implies σ ( w ) = w , and hence σ = id logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F = µ ( w ) r k = wr j r k = wr j + k � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ and hence Φ ( µ ◦ σ ) = [ j + k ] n = [ j ] n +[ k ] n = Φ ( µ )+ Φ ( σ ) . Moreover, Φ is injective, because Φ ( σ ) = [ 0 ] n implies σ ( w ) = w , and hence σ = id (rest: good exercise). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F = µ ( w ) r k = wr j r k = wr j + k � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ and hence Φ ( µ ◦ σ ) = [ j + k ] n = [ j ] n +[ k ] n = Φ ( µ )+ Φ ( σ ) . Moreover, Φ is injective, because Φ ( σ ) = [ 0 ] n implies σ ( w ) = w , and hence σ = id (rest: good exercise). � � F ( w ) / F Hence G is isomorphic to a subgroup of Z n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F = µ ( w ) r k = wr j r k = wr j + k � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ and hence Φ ( µ ◦ σ ) = [ j + k ] n = [ j ] n +[ k ] n = Φ ( µ )+ Φ ( σ ) . Moreover, Φ is injective, because Φ ( σ ) = [ 0 ] n implies σ ( w ) = w , and hence σ = id (rest: good exercise). � � F ( w ) / F Hence G is isomorphic to a subgroup of Z n , and because Z n is commutative, so are its subgroups. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let r be a primitive n th root of unity. Then 1 , r , r 2 ,..., r n − 1 are the n distinct n th roots of unity and w , rw , r 2 w ,..., r n − 1 w are the n distinct roots of x n − f . Therefore F ( w ) is the splitting field of the polynomial x n − f . . Then σ ( w ) = wr k for some k ∈ Z , which � � Let σ ∈ G F ( w ) / F � � is unique modulo n . Define Φ : G F ( w ) / F → Z n by Φ ( σ ) : = [ k ] n , where k is the unique element of { 0 ,..., n − 1 } so that σ ( w ) = wr k . and let µ ( w ) = wr j , σ ( w ) = wr k . Then � � Let σ , µ ∈ G F ( w ) / F = µ ( w ) r k = wr j r k = wr j + k � wr k � � r k � µ ◦ σ ( w ) = µ = µ ( w ) µ and hence Φ ( µ ◦ σ ) = [ j + k ] n = [ j ] n +[ k ] n = Φ ( µ )+ Φ ( σ ) . Moreover, Φ is injective, because Φ ( σ ) = [ 0 ] n implies σ ( w ) = w , and hence σ = id (rest: good exercise). � � F ( w ) / F Hence G is isomorphic to a subgroup of Z n , and because Z n is commutative, so are its subgroups. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Definition. Let G be a finite group. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Definition. Let G be a finite group. Then G is called solvable iff there is a chain of subgroups { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G so that G j / G j − 1 is commutative for j = 1 ,..., m. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Definition. Let G be a finite group. Then G is called solvable iff there is a chain of subgroups { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G so that G j / G j − 1 is commutative for j = 1 ,..., m. Lemma. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Definition. Let G be a finite group. Then G is called solvable iff there is a chain of subgroups { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G so that G j / G j − 1 is commutative for j = 1 ,..., m. Lemma. Let G be a finite group and let N ⊳ G . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Definition. Let G be a finite group. Then G is called solvable iff there is a chain of subgroups { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G so that G j / G j − 1 is commutative for j = 1 ,..., m. Lemma. Let G be a finite group and let N ⊳ G . If G is solvable, then G / N is solvable. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 x ˜ n x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN = g j N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN = g j N . So each element of G j N / N is of the form g j N for some g j ∈ G j . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN = g j N . So each element of G j N / N is of the form g j N for some g j ∈ G j . G j − 1 N / N is normal in G j N / N: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN = g j N . So each element of G j N / N is of the form g j N for some g j ∈ G j . G j − 1 N / N is normal in G j N / N: G j − 1 N / N is a subgroup of G j N / N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN = g j N . So each element of G j N / N is of the form g j N for some g j ∈ G j . G j − 1 N / N is normal in G j N / N: G j − 1 N / N is a subgroup of G j N / N . Let g j N ∈ G j N / N and let g j − 1 N ∈ G j − 1 N / N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN = g j N . So each element of G j N / N is of the form g j N for some g j ∈ G j . G j − 1 N / N is normal in G j N / N: G j − 1 N / N is a subgroup of G j N / N . Let g j N ∈ G j N / N and let g j − 1 N ∈ G j − 1 N / N . Then ( g j N )( g j − 1 N ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN = g j N . So each element of G j N / N is of the form g j N for some g j ∈ G j . G j − 1 N / N is normal in G j N / N: G j − 1 N / N is a subgroup of G j N / N . Let g j N ∈ G j N / N and let g j − 1 N ∈ G j − 1 N / N . Then ( g j N )( g j − 1 N ) = Ng j g j − 1 N logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN = g j N . So each element of G j N / N is of the form g j N for some g j ∈ G j . G j − 1 N / N is normal in G j N / N: G j − 1 N / N is a subgroup of G j N / N . Let g j N ∈ G j N / N and let g j − 1 N ∈ G j − 1 N / N . Then ( g j N )( g j − 1 N ) = Ng j g j − 1 N = Ng ′ j − 1 g j N logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN = g j N . So each element of G j N / N is of the form g j N for some g j ∈ G j . G j − 1 N / N is normal in G j N / N: G j − 1 N / N is a subgroup of G j N / N . Let g j N ∈ G j N / N and let g j − 1 N ∈ G j − 1 N / N . Then � � ( g j N )( g j − 1 N ) = Ng j g j − 1 N = Ng ′ g ′ j − 1 g j N = ( g j N ) j − 1 N logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN = g j N . So each element of G j N / N is of the form g j N for some g j ∈ G j . G j − 1 N / N is normal in G j N / N: G j − 1 N / N is a subgroup of G j N / N . Let g j N ∈ G j N / N and let g j − 1 N ∈ G j − 1 N / N . Then � � ( g j N )( g j − 1 N ) = Ng j g j − 1 N = Ng ′ g ′ j − 1 g j N = ( g j N ) , j − 1 N which proves that ( g j N )( G j − 1 N / N ) ⊆ ( G j − 1 N / N )( g j N ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN = g j N . So each element of G j N / N is of the form g j N for some g j ∈ G j . G j − 1 N / N is normal in G j N / N: G j − 1 N / N is a subgroup of G j N / N . Let g j N ∈ G j N / N and let g j − 1 N ∈ G j − 1 N / N . Then � � ( g j N )( g j − 1 N ) = Ng j g j − 1 N = Ng ′ g ′ j − 1 g j N = ( g j N ) , j − 1 N which proves that ( g j N )( G j − 1 N / N ) ⊆ ( G j − 1 N / N )( g j N ) . Thus for all x ∈ G j N / N we have x ( G j − 1 N / N ) ⊆ ( G j − 1 N / N ) x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN = g j N . So each element of G j N / N is of the form g j N for some g j ∈ G j . G j − 1 N / N is normal in G j N / N: G j − 1 N / N is a subgroup of G j N / N . Let g j N ∈ G j N / N and let g j − 1 N ∈ G j − 1 N / N . Then � � ( g j N )( g j − 1 N ) = Ng j g j − 1 N = Ng ′ g ′ j − 1 g j N = ( g j N ) , j − 1 N which proves that ( g j N )( G j − 1 N / N ) ⊆ ( G j − 1 N / N )( g j N ) . Thus for all x ∈ G j N / N we have x ( G j − 1 N / N ) ⊆ ( G j − 1 N / N ) x , that is, x ( G j − 1 N / N ) x − 1 ⊆ ( G j − 1 N / N ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof. Let { e } = G 0 ⊳ G 1 ⊳ ··· ⊳ G m = G be a chain of subgroups so that G j / G j − 1 is commutative for j = 1 ,..., m . Each G j N is a subgroup of G: e ∈ G j N � = / 0. Let x , y ∈ G j N . Then there are g x , g y ∈ G j and n x , n y ∈ N so that x = g x n x and y = g y n y . Now xy = ( g x n x )( g y n y ) = g x ( n x g y ) n y = g x ( g y n ) n y = ( g x g y )( nn y ) ∈ G j N and x − 1 = n − 1 x g − 1 = g − 1 n ∈ G j N . x ˜ x N ⊳ G , so N ⊳ G j N . If xN ∈ G j N / N , then there are a g j ∈ G j and an n ∈ N so that xN = g j nN = g j N . So each element of G j N / N is of the form g j N for some g j ∈ G j . G j − 1 N / N is normal in G j N / N: G j − 1 N / N is a subgroup of G j N / N . Let g j N ∈ G j N / N and let g j − 1 N ∈ G j − 1 N / N . Then � � ( g j N )( g j − 1 N ) = Ng j g j − 1 N = Ng ′ g ′ j − 1 g j N = ( g j N ) , j − 1 N which proves that ( g j N )( G j − 1 N / N ) ⊆ ( G j − 1 N / N )( g j N ) . Thus for all x ∈ G j N / N we have x ( G j − 1 N / N ) ⊆ ( G j − 1 N / N ) x , that is, x ( G j − 1 N / N ) x − 1 ⊆ ( G j − 1 N / N ) and G j − 1 N / N ⊳ G j N / N . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). ( G j N / N ) / ( G j − 1 N / N ) is commutative: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). ( G j N / N ) / ( G j − 1 N / N ) is commutative: The elements of ( G j N / N ) / ( G j − 1 N / N ) are of the form g j NG j − 1 N with g j ∈ G j . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). ( G j N / N ) / ( G j − 1 N / N ) is commutative: The elements of ( G j N / N ) / ( G j − 1 N / N ) are of the form g j NG j − 1 N with g j ∈ G j . Hence each element of ( G j N / N ) / ( G j − 1 N / N ) is of the form g j NG j − 1 N logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). ( G j N / N ) / ( G j − 1 N / N ) is commutative: The elements of ( G j N / N ) / ( G j − 1 N / N ) are of the form g j NG j − 1 N with g j ∈ G j . Hence each element of ( G j N / N ) / ( G j − 1 N / N ) is of the form g j NG j − 1 N = g j G j − 1 NN logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). ( G j N / N ) / ( G j − 1 N / N ) is commutative: The elements of ( G j N / N ) / ( G j − 1 N / N ) are of the form g j NG j − 1 N with g j ∈ G j . Hence each element of ( G j N / N ) / ( G j − 1 N / N ) is of the form g j NG j − 1 N = g j G j − 1 NN = g j G j − 1 N logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). ( G j N / N ) / ( G j − 1 N / N ) is commutative: The elements of ( G j N / N ) / ( G j − 1 N / N ) are of the form g j NG j − 1 N with g j ∈ G j . Hence each element of ( G j N / N ) / ( G j − 1 N / N ) is of the form g j NG j − 1 N = g j G j − 1 NN = g j G j − 1 N with g j ∈ G j . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). ( G j N / N ) / ( G j − 1 N / N ) is commutative: The elements of ( G j N / N ) / ( G j − 1 N / N ) are of the form g j NG j − 1 N with g j ∈ G j . Hence each element of ( G j N / N ) / ( G j − 1 N / N ) is of the form g j NG j − 1 N = g j G j − 1 NN = g j G j − 1 N with g j ∈ G j . Now let g j G j − 1 N , h j G j − 1 N ∈ ( G j N / N ) / ( G j − 1 N / N ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). ( G j N / N ) / ( G j − 1 N / N ) is commutative: The elements of ( G j N / N ) / ( G j − 1 N / N ) are of the form g j NG j − 1 N with g j ∈ G j . Hence each element of ( G j N / N ) / ( G j − 1 N / N ) is of the form g j NG j − 1 N = g j G j − 1 NN = g j G j − 1 N with g j ∈ G j . Now let g j G j − 1 N , h j G j − 1 N ∈ ( G j N / N ) / ( G j − 1 N / N ) . Then ( g j G j − 1 N )( h j G j − 1 N ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). ( G j N / N ) / ( G j − 1 N / N ) is commutative: The elements of ( G j N / N ) / ( G j − 1 N / N ) are of the form g j NG j − 1 N with g j ∈ G j . Hence each element of ( G j N / N ) / ( G j − 1 N / N ) is of the form g j NG j − 1 N = g j G j − 1 NN = g j G j − 1 N with g j ∈ G j . Now let g j G j − 1 N , h j G j − 1 N ∈ ( G j N / N ) / ( G j − 1 N / N ) . Then ( g j G j − 1 N )( h j G j − 1 N ) = g j G j − 1 h j NG j − 1 N logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). ( G j N / N ) / ( G j − 1 N / N ) is commutative: The elements of ( G j N / N ) / ( G j − 1 N / N ) are of the form g j NG j − 1 N with g j ∈ G j . Hence each element of ( G j N / N ) / ( G j − 1 N / N ) is of the form g j NG j − 1 N = g j G j − 1 NN = g j G j − 1 N with g j ∈ G j . Now let g j G j − 1 N , h j G j − 1 N ∈ ( G j N / N ) / ( G j − 1 N / N ) . Then ( g j G j − 1 N )( h j G j − 1 N ) = g j G j − 1 h j NG j − 1 N = g j G j − 1 h j G j − 1 NN logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). ( G j N / N ) / ( G j − 1 N / N ) is commutative: The elements of ( G j N / N ) / ( G j − 1 N / N ) are of the form g j NG j − 1 N with g j ∈ G j . Hence each element of ( G j N / N ) / ( G j − 1 N / N ) is of the form g j NG j − 1 N = g j G j − 1 NN = g j G j − 1 N with g j ∈ G j . Now let g j G j − 1 N , h j G j − 1 N ∈ ( G j N / N ) / ( G j − 1 N / N ) . Then ( g j G j − 1 N )( h j G j − 1 N ) = g j G j − 1 h j NG j − 1 N = g j G j − 1 h j G j − 1 NN = h j G j − 1 g j G j − 1 NN logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). ( G j N / N ) / ( G j − 1 N / N ) is commutative: The elements of ( G j N / N ) / ( G j − 1 N / N ) are of the form g j NG j − 1 N with g j ∈ G j . Hence each element of ( G j N / N ) / ( G j − 1 N / N ) is of the form g j NG j − 1 N = g j G j − 1 NN = g j G j − 1 N with g j ∈ G j . Now let g j G j − 1 N , h j G j − 1 N ∈ ( G j N / N ) / ( G j − 1 N / N ) . Then ( g j G j − 1 N )( h j G j − 1 N ) = g j G j − 1 h j NG j − 1 N = g j G j − 1 h j G j − 1 NN = h j G j − 1 g j G j − 1 NN = h j G j − 1 g j NG j − 1 N logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups Proof (concl.). ( G j N / N ) / ( G j − 1 N / N ) is commutative: The elements of ( G j N / N ) / ( G j − 1 N / N ) are of the form g j NG j − 1 N with g j ∈ G j . Hence each element of ( G j N / N ) / ( G j − 1 N / N ) is of the form g j NG j − 1 N = g j G j − 1 NN = g j G j − 1 N with g j ∈ G j . Now let g j G j − 1 N , h j G j − 1 N ∈ ( G j N / N ) / ( G j − 1 N / N ) . Then ( g j G j − 1 N )( h j G j − 1 N ) = g j G j − 1 h j NG j − 1 N = g j G j − 1 h j G j − 1 NN = h j G j − 1 g j G j − 1 NN = h j G j − 1 g j NG j − 1 N = ( h j G j − 1 N )( g j G j − 1 N ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals