Constructing a Finite Field of Order 16 and Finding Primative Element

In summary, the conversation discusses constructing a finite field of order 16 and finding a primitive element. The solution involves finding an irreducible polynomial and setting a root of it to be the primitive element. While this method may work for some fields, it is not always the case and finding the primitive element can be a difficult problem.
  • #1
fireisland27
11
0

Homework Statement



Construct a finite field of order 16. And find a primative element.

Homework Equations





The Attempt at a Solution



What I did was find an irreducible polynomial in Z/<2> of degree 4. I used f(x)=x^4+x+1.
Then I took a to be a root of f(x) and set a^4=a+1. Then to make the field I just took powers of a. a is clearly a primitive element.
This seems too easy. Does this indeed produce the field? And does this exact method work for constructing any finite field? And if so doesn't it always give us a primitive element right away?
 
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  • #2
Yes, I believe this solution is correct.

However, finding the primitive element is in general much harder. If P(X) is an irreducible polynomial, and if you're working in the field [tex]\mathbb{F}_p[X]/(P(X))[/tex], then it is not always the case that X is the primitive element. In fact, finding the primitive element of a finite field is quite a difficult programming problem nowadays, certainly for big fields...
 
  • #3
What would be an example where x is not a primitive element?
 
  • #4
For example [tex]\mathbb{F}_3[X]/(X^2+1)[/tex] is a field of 9 elements. But the generator is X+1, rather then X...
 

1. How do you construct a finite field of order 16?

To construct a finite field of order 16, we start with a prime number p and find the smallest positive integer n such that p^n - 1 is divisible by 16. We then take the polynomial f(x) = x^n - 1 and find its irreducible factors over the field of integers modulo p. These factors will be the basis of the finite field of order 16.

2. What is a primitive element in a finite field?

A primitive element in a finite field is an element that generates the multiplicative group of the field. This means that every non-zero element in the field can be expressed as a power of the primitive element. In other words, the primitive element is a generator of the field.

3. How do you find a primitive element in a finite field of order 16?

To find a primitive element in a finite field of order 16, we first construct the field using the steps mentioned in the answer to the first question. Then, we test each element in the field to see if it generates the multiplicative group. If an element does not generate the group, we move on to the next one. This process continues until we find a primitive element.

4. What is the significance of finding a primitive element in a finite field of order 16?

Finding a primitive element in a finite field of order 16 is significant because it allows us to perform various operations in the field efficiently. For example, finding the inverse of an element or raising an element to a power can be done more quickly using a primitive element.

5. Can there be multiple primitive elements in a finite field of order 16?

Yes, there can be multiple primitive elements in a finite field of order 16. In fact, the number of primitive elements is equal to the number of generators of the multiplicative group, which is equal to the number of elements in the group minus 1. In the case of a finite field of order 16, there will be 15 primitive elements.

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