Help me construcing finite field,

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To construct a finite field with q elements where 10^9 < q < 10^10, a prime number p and an integer n must be chosen such that p^n falls within this range. The construction involves using the field Z_p and an irreducible polynomial of degree n. Only one irreducible polynomial is necessary for the construction, not all possible irreducible ones. Tutorials on finite fields can provide additional guidance on the process. Understanding the basics of finite fields and irreducible polynomials is essential for successful construction.
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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?
 
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What do you know about fields?
Do you have any examples of fields?
 
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
 
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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?
 
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