Re: Relationship between polynomial of Galois group "[2^4]E(4)" and a polynomial of degree 16.

On Nov 21, 6:30 pm, Gerry <Gerry...@xxxxxxxxx> wrote:
Hi all,

the irreducible polynomial P1 (see below) of degree 8 has Galois
group :"[2^4]E(4)".
What is the Galois group of the irreucible polynomial P2 of degree 16?
Is there a relationship between the two Galois groups?

 P1= a8*x^8 + a7*x^7 + a6*x^6 + a5*x^5 + a4*x^4 + a3*x^3 + a2*x^2 +
a1*x^1 + a0

 coefficients of P1 with b,c in Q :

 a0= 1
 a1= 0
 a2= 0
 a3= 0
 a4=     c^4
 a5= 4* c^3 * x
 a6= 6* c^2 * x^2
 a7= 4* c   *  x^3
 a8=             x^4

 P2= a16*x^16 + a15*x^15 + a14*x^14 + a13*x^13 + a12*x^12 + a11*x^11 +
a10*x^10 + a9*x^9
    + a8*x^8 + a7*x^7 + a6*x^6 + a5*x^5 + a4*x^4 + a3*x^3 + a2*x^2 +
a1*x^1 + a0

 coefficients of P2 with b,c in Q :

 a0=   1
 a1=   0
 a2=   0
 a3=   0
 a4= -      c^4
 a5= - 4 * c^3   * b
 a6= - 6 * c^2   * b^2
 a7= - 4 * c      * b^3
 a8=          c^8 - b^4
 a9=     8 * c^7 * b
 a10= 28 * c^6 * b^2
 a11= 56 * c^5 * b^3
 a12= 70 * c^4 * b^4
 a13= 56 * c^3 * b^5
 a14= 28 * c^2 * b^6
 a15=   8 * c   *  b^7
 a16=                b^8

Regards

Gerry

Correction

coefficients of P1 with b,c in Q :

The P1 coefficients should be:

a0= 1
a1= 0
a2= 0
a3= 0
a4= c^4
a5= 4* c^3 * b
a6= 6* c^2 * b^2
a7= 4* c * b^3
a8= b^4


.