- #1

PsychonautQQ

- 784

- 10

## Homework Statement

Find the Galois group of f(x) = x^7-x^6-2x+2 over ##F_7##.

## Homework Equations

## The Attempt at a Solution

1 is a root of f(x) so dividing f(x) / (x-1) we get the quotient x^6-2. Now all elements of ##F_7## satisfy a^6 = 1 since it's multiplicative group is of order 6, and thus no elements will be a root of x^6-2. Thus I will add an element r such that r^6 = 2 to ##F_7## and now a*(r^6) will be roots for all a in ##F_7## and the splitting field is Q(r).

As far as Gal(Q(r):Q) goes, any automorphism in this group will fix all of Q, and thus the only thing it can move is r and it can only go to another root of it's minimal polynomial (x^6-2), so I believe there is a unique automorphism that will for each element of ##F_7##, where 1(r)=r, 2(r)=2r, 3(r)=3r, etc. Thus the structure of the Galois group will be isomorphic to the multiplicative group of units in ##F_7##.

Is this correct?