Formal Derivative and Multiple Roots

[JonoPUH]

Primitive roots of 1 over a finite field

Homework Statement

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×.

Homework Equations

The Attempt at a Solution

I am totally confused about how to even start this proof. Any help would be greatly appreciated please!

 

[mfb]

β³-2=0
or, equivalent, β³+5=0
You can add and multiply polynomials with β as usual, just consider everything mod 7, and use that equation to keep the degree of your polynomials smaller than 3.

Therefore, -1 = 6 (as 6+1=0), and 6*6=...
You can calculate β, β^2 (well, nothing to do for those), β^3, β^4, ... and show that it is a primitive 9th root in that way.
(3) would be possible like that, too, but there might be a more elegant way.