I don't know how to solve these exercises.
Let be a field containing m elements and be positive.
Let . Show:
Let g: , . Find all elements of that map to
Notes: Those fields are finite at least are we doing this topic actually. m does not have to be a prime we have to do the proof for 1) in general.
And thats my problem for exercise 1)
If . The order of x is a divisor of m ? At least is true? I don't understand why this is true at this point already... If m is not prime (and we can't assume that) the number of units in doesn't have to be m-1 yeah???
If so the equation could be multiplied by x and what results is after subtracting x on both sides and this direction is proved.
Here from follows that so by dividing by x we have thus x has order m-1 so it is contained in the subset which consists of elements with order m-1 ????
Let's have a look at exercise 2)
So is isomorphic to where ?
...have no idea
Do you know how this works? Could you please give me a hint?