Constructing a finite field with 10^9 to 10^10 elements in Mathematica

[Granit_niti]

Help me construcing finite field, please!

I need Construct a finite field with q Elements (10^9<q<10^10) and find
a primitive root. That should be done in mathematica but I should not use Package for Finite Fields.

Any idea how to solve this problem?

 

[NateTG]

What do you know about fields?
Do you have any examples of fields?

 

[JonF]

I don’t think this is pre-calc, but:

Can you think of a number p and n where

10^9 < p^n < 10^10

you should know how to construct a field with p^n elements.

Hint: It involves Z_p and a irreducible polynomial of order n (check your class notes!)


Edit: p must be prime of course

 

[Granit_niti]

Thank you very much!

Could you refer me to any tutorial for constructing field with p^n elements the?
I am not good with fields at all!

When I construct the field do I have to take all polynomials possible of n degree in Z/Zp or only one irreducible or all irreducible ones?

I don't have clear what is the finite field. What do I have to find in order to construct finite field?