Originally Posted by
knguyen2005 I have been given this question:
Find Aut (F_2[x]/(x^3+x+1)) ?
I can do this question easily if the question asked: Find Aut (F_2[x]/(x^3+x+1))* where (F_2[x]/(x^3+x+1))* denotes the group
This is my work:
F_2 = {0,1}
Set x^3 + x + 1 = 0 then x^3 = -x -1 = x + 1
There are 8 elements in F_2[x]/(x^3+x+1): 0, 1, x, x+1, x^2, x^2 +x, x^2 +1, x^2 +x +1
Since x^3 + x + 1 is irreducible over F_2 , so (F_2[x]/(x^3+x+1)) is a field
If we exclude 0 then (F_2[x]/(x^3+x+1)) becomes a group, and denote this by (F_2[x]/(x^3+x+1))* = G
Since org (G) = |(F_2[x]/(x^3+x+1))*| = 7 then we conclude that
(F_2[x]/(x^3+x+1))* =~ C_7 where x is the generator of G
So: Aut (F_2[x]/(x^3+x+1))* =~ Aut(C_7) = C_6 (since 7 is prime)
BUT in this case the question is different. So I am wondering if Aut(R) and Aut(G) is the same? ( R = ring and G =group)
If I am wrong then, how do you solve this question.
Thank you for reading my thread.