- #1

PsychonautQQ

- 784

- 10

## Homework Statement

2 questions here:

1)

Let g(x) = x^6 - 10 be a polynomial in Q(c) where c is a primitive 6th root of unity. Find a splitting field for this polynomial and determine it's Galois group

2)

let f(x) = x^3 + x^2 + 2 with coefficients in ##F_3##. Find a splitting field K for this polynomial and determine it's Galois group

## Homework Equations

## The Attempt at a Solution

Question 1)

If we adjoin b = the 6th real root of 10 to Q(c) then we have a splitting field. Furthermore, the roots of f will be {b, bc, bc^2, bc^3, bc^4, bc^5}. The Galois group will permute the roots whilst fixing every element of Q(c). So I believe this galois group will be isomorphic to ##(Z_6,+)## where the generator will take b-->bc and fix all elements of Q(c).

Question 2)

I'm having trouble constructing a splitting field for this polynomial. If I assume r is a root and then divide f(x) by x-r I don't seem to arrive at anything helpful. Perhaps I should consider that the multiplicative group in F_7(r) will be cyclic of order 26 (since the degree of r is 3 and 3*9 = 27) so r^27=r, and then use this knowledge to see if any powers of r are also roots. Does this make sense? Does anyone else have any other insights to make this simpler?