Primitive roots of 1 over a finite field 1. The problem statement, all variables and given/known data The polynomial x3 − 2 has no roots in F7 and is therefore irreducible in F7[x]. Adjoin a root β to make the field F := F7(β), which will be of degree 3 over F7 and therefore of size 343. The multiplicative group F× is of order 2 × 9 × 19. Show: (1) that −1 is a primitive square root of 1 in F; (2) that β is a primitive 9th root of 1 in F; (3) that −1 + β is a primitive 19th root of 1 in F; (4) and then that β − β2 is a primitive root in F, that is, a generator of the multiplicative group F×. 2. Relevant equations 3. The attempt at a solution I am totally confused about how to even start this proof. Any help would be greatly appreciated please!