1.
Compute the conjugacy classes of the elements of \(E\text{.}\)
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Worksheet Properties of Minimal Polynomials & Finite Field Isomorphisms
Today we’ll see more properties of minimal polynomials of finite field elements and actually prove our claim from early in the semester that there is only one finite field structure for each possible order of finite field.
In the next series of problems, you will investigate the structure of a field \(E\) of order \(2^4\text{,}\) constructed as \(GF(2)[\alpha]/(\alpha^4+\alpha+1)\text{.}\)
2.
List all of the irreducible polynomials over \(GF(2)\) of degree \(4\) or less. (You’ve done this computation before!)
3.
What sizes of subfields can \(E\) have?
4.
Verify that the product of the linear and quadratic irreducible polynomials is \(x^4-x\) and use this to identify the subfield \(K\) of \(E\) of size \(4\text{.}\)
5.
Match the remaining elements of \(E\) with their minimal polynomials. What degree is each minimal polynomial?
6.
Draw a diagram showing the subfields of \(E\text{.}\) Include labels for which field is which and which conjugacy classes live in which parts of the diagram.
In the next series of problems you will investigate the structure of a field \(K\) of order \(2^6\) with primitive element \(\beta\)
7.
What sizes of subfields can \(K\) have?
8.
What degrees can minimal polynomials of elements of \(K\) have?
9.
Identify the minimal polynomials which are factors of \(x^4-x\) and corresponding elements (as powers of \(\beta\)) that comprise the subfield of size \(4\) in \(K\text{.}\)
10.
Identify the minimal polynomials which are factors of \(x^8-x\) and corresponding elements (as powers of \(\beta\)) that comprise the subfield of size \(8\) in \(K\text{.}\)
11.
What is the intersection of these two subfields?
12.
Based on your work above, how many elements of \(K\) have minimal polynomials of degree \(6\text{?}\) What does this tell you about the number of monic irreducible polynomials of degree \(6\) over \(\GF(2)\text{?}\)
