Find the splitting field of x6 1 over zopf2 and the corresp

Find the splitting field of x^6 + 1 over zopf_2, and the corresponding Galois group. Carefully explain your thoughts.

Solution

We know that:

x⁶ + 1 = (x³ + 1)²

so it is sufficient to find a splitting field for x³ + 1.

Also, x³ + 1 = (x + 1)(x 2 x + 1)

The polynomial x² x + 1 is irreducible over Z₂, because

0² 0 + 1 = 1² 1 + 1 = 1 != 0,

which shows it has no roots in Z₂.

Let be a root of x² + x + 1 in some field extension of Z₂.

Compute that:

( + 1)² ( + 1) + 1 = ² + 1 1 + 1 = ² + 1 = 0

Thus, it’s clear that: x⁶ + 1 = (x 1)²(x )²(x 1)²

Hence the splitting field for x⁶ + 1 is Z₂[], which has degree 2