Thread: finite fields

Let F=(Z₃[x])/<x²+1>
Calculate (x + 1)⁴ in F. Explain why your calculation shows that F* is cyclic

I have calculated (x + 1)⁴and then divided the result by x^2+1 which gives me x+2

I am not sure if i have done this correctly and i dont see how this shows me that the multiplicative group will be cyclic.

Originally Posted by ulysses123

Let F=(Z₃[x])/<x²+1>
Calculate (x + 1)⁴ in F. Explain why your calculation shows that F* is cyclic

I have calculated (x + 1)⁴and then divided the result by x^2+1 which gives me x+2

I am not sure if i have done this correctly and i dont see how this shows me that the multiplicative group will be cyclic.

It seems you wrote the field with 9 elements.
Now , I don't know what is or whatever it was you wrote, but perhaps the intention was to

calculate the powers of and, thus, show that this is an

element of order 8, thus showing that is a cyclic group...
But then I don't understand why did you divide ...??

Explain better your symbols and what exactly have you done.

Anyway, it's easy to check that and thus indeed has order 8 in the multiplicative group of the field.

Tonio