Help me construcing finite field,

[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?

 

[morphism]

What do you mean by primitive root here?

 

[tacman]

Sure, I can help you construct a finite field with q elements and find a primitive root. First, let's define some terms:

- A finite field is a mathematical structure that consists of a finite set of elements and two operations, addition and multiplication. The number of elements in a finite field is known as the order of the field.
- A primitive root of a finite field is an element that generates all the non-zero elements of the field when raised to different powers.

To construct a finite field with q elements, we can use the following steps:

1. Determine the order of the field, q, which is a prime number between 10^9 and 10^10.
2. Choose a primitive element, g, which will be our primitive root. This element must be a primitive root modulo q, meaning that it must generate all the non-zero elements of the field when raised to different powers.
3. Create a set of elements, S, by taking all the powers of g modulo q. This set will contain q-1 elements and will be the elements of our finite field.
4. Define addition and multiplication operations on S, such that they follow the rules of a finite field. For example, addition will be defined as a + b = (a + b) mod q and multiplication as a * b = (a * b) mod q.
5. Verify that the operations defined in step 4 satisfy all the properties of a finite field, such as associativity, commutativity, and distributivity.
6. Your finite field with q elements is now constructed.

To find a primitive root, you can use the following steps:

1. Calculate the Euler totient function, φ(q), which is the number of positive integers less than q that are relatively prime to q. This can be done using the formula φ(q) = q-1.
2. Choose a random element, g, from the set of positive integers less than q.
3. Raise g to the power of φ(q)/p for all prime factors p of φ(q). If g^(φ(q)/p) mod q = 1 for any prime factor p, then g is not a primitive root.
4. If g^(φ(q)/p) mod q ≠ 1 for all prime factors p of φ(q), then g is a primitive root of the finite field with q elements.

I hope this helps you construct a finite field and find a primitive root without