Exercise 13.4.11

Proof.

(a)

There are five transitive subgroups of $S_4$ - $\{S_4,A_4,C_2\times C_2,C_4,D_8\}$. The following Sage instructions produce the similar to (13.46) table with the percentage of elements of each possible cycle for each subgroup:

     S4 = SymmetricGroup(4)
     A4 = AlternatingGroup(4)
     C2C2 = PermutationGroup(["(1,3)(2,4)","(1,2)(3,4)"])
     C4 = CyclicPermutationGroup(4)
     D8 = DihedralGroup(4)
     G =[S4,A4,C2C2,C4,D8]
     
     for j in range(5):
         V =[g.cycle_type() for g in G[j]]
         K=[0,0,0,0,0]
         for i in range(len(V)):
             if V[i]==[1,1,1,1] : K[0]=K[i]+1
             elif V[i]==[2,1,1] : K[1]=K[1]+1
             elif V[i]==[3,1] : K[2]=K[2]+1
             elif V[i]==[2,2] : K[3]=K[3]+1
             elif V[i]==[4] : K[4]=K[4]+1
             else : print("Error")
             Sm=sum(K)
         if j==0 :
             row = matrix([round(K[0]/Sm*100.0,1), round(K[1]/Sm*100.0,1)
                 ,round(K[2]/Sm*100.0,1)*1,round(K[3]/Sm*100.0,1)
                 ,round(K[4]/Sm*100.0,1)])
         else :
             U = [round(K[0]/Sm*100.0,1), round(K[1]/Sm*100.0,1)
                 ,round(K[2]/Sm*100.0,1)*1,round(K[3]/Sm*100.0,1)
                 ,round(K[4]/Sm*100.0,1)]]
             row = matrix(row.rows()+[U])
     C=[["1,1,1,1","1,1,2","1,3","2,2","4"]]+[list(row) for row in row]
     A = ["cycle type","S_4","A_4","C_2xC_2","C_4","D_8"]
     C = [[a] + u for a, u in zip(A, C)]
     (table(C, header_row=True, frame=True))

cycle type	1,1,1,1	1,1,2	1,3	2,2	4
$S_4$	$4.2$	$25.0$	$33.3$	$12.5$	$25.0$
$A_4$	$8.3$	$0.0$	$66.7$	$25.0$	$0.0$
$C_2\times C_2$	$25.0$	$0.0$	$0.0$	$75.0$	$0.0$
$C_4$	$25.0$	$0.0$	$0.0$	$25.0$	$50.0$
$D_8$	$12.5$	$25.0$	$0.0$	$37.5$	$25.0$

(b)

The following Sage instructions produce the similar to (13.45) table. The percentage of primes corresponding to each possible cycle for 200 primes $7\le p\le 1237$ is calculated for each polynomial in Exercise 13.1.14 and the polynomial from Example 13.4.8:

     R.<X> = PolynomialRing(ZZ)
     P = Primes()
     f=[X^4 - 7*X^3 + 19*X^2 - 23*X + 11,X^4 + 4*X + 2,X^4 + 8*X + 12
         ,X^4 + 1,X^4 + X^3 + X^2 + X + 1,X^4 - 2]
     for r in range(6):
         K=[0,0,0,0,0]
         for i in range(2,203):
             R11.<x> = PolynomialRing(GF(P.unrank(i)))
             S=[0,0,0,0]
             for j in range(len(R11(f[r]).factor())):
                 d=R11(f[r]).factor()[j][0].degree()
                 m=R11(f[r]).factor()[j][1]
                 S[d-1]=S[d-1]+m
             if S[0]==4: K[0]=K[0]+1
             elif S[0]==2 and S[1]==1: K[1]=K[1]+1
             elif S[0]==1 and S[2]==1: K[2]=K[2]+1
             elif S[1]==2 : K[3]=K[3]+1
             elif S[3]==1 : K[4]=K[4]+1
             else : print("Error")
             Sm=sum(K)
         if r==0 :
             row = matrix([round(K[0]/Sm*100.0,1), round(K[1]/Sm*100.0,1)
                 ,round(K[2]/Sm*100.0,1)*1,round(K[3]/Sm*100.0,1)
                 ,round(K[4]/Sm*100.0,1)])
         else :
             U = [round(K[0]/Sm*100.0,1), round(K[1]/Sm*100.0,1)
                 ,round(K[2]/Sm*100.0,1)*1,round(K[3]/Sm*100.0,1)
                 ,round(K[4]/Sm*100.0,1)]
             row = matrix(row.rows()+[U])
     C=[["1,1,1,1","1,1,2","1,3","2,2","4"]]+[list(row) for row in row]
     A = ["cycle type"]+f
     C = [[a] + u for a, u in zip(A, C)]
     B = ["Galois group","C_4","S_4","A_4","C_2xC_2","C_4","D_8"]
     C = [a + [u] for a, u in zip(C, B)]
     (table(C, header_row=True, frame=True))

cycle type	1,1,1,1	1,1,2	1,3	2,2	4	Galois group
$X^4-7X^3+19X^2-23X+11$	$25.4$	$0.0$	$0.0$	$22.9$	$51.7$	$C_4$
$X^4+4X+2$	$3.0$	$26.9$	$33.8$	$13.4$	$22.9$	$S_4$
$X^4+8X+12$	$6.5$	$0.0$	$67.2$	$26.4$	$0.0$	$A_4$
$X^4+1$	$22.9$	$0.0$	$0.0$	$77.1$	$0.0$	$C_2\times C_2$
$X^4+X^3+X^2+X+1$	$25.4$	$0.0$	$0.0$	$22.9$	$51.7$	$C_4$
$X^4-2$	$10.4$	$25.4$	$0.0$	$37.3$	$26.9$	$D_8$

The last column provides the guess about the Galois group of each polynomial. The comparison with Exercise 14 of Section 13.1 and Example 13.4.8 results supports the Chebotarev Density Theorem.