1.
True/False: The coefficients of the minimal polynomial with respect to \(F\) of \(\alpha\) a nonzero element of an extension field \(E\) of \(F\) are elements of \(F\text{.}\)
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Worksheet Weekly Practice 12
Instructions: You may type up or handwrite your work, but it must be neat, professional, and organized and it must be saved as a PDF file and uploaded to the appropriate Gradescope assignment. Use a scanner or scanning app to convert handwritten work on paper to PDF. I encourage you to type your work using the provided template.
All tasks below must have a complete solution that represents a good-faith attempt at being right to receive engagement credits. If your submission is complete and turned in on time, you will receive full engagement credit for the assignment. All other submissions will receive zero engagement credit. Read the guidelines at Grading Specifications carefully.
To abide by the class academic honesty policy, your work should represent your own understanding in your own words. If you work with other students, you must clearly indicate who you worked with in your submission. The same is true for using tools like generative AI although I strongly discourage you from using such tools since you need to build your own understanding here to do well on exams.
True/False, Multiple Choice, & Fill-In.
For these problems a justification is not required for credit, but it may be useful for your own understanding to include one. True/False problems should be marked True if the statement is always true, and False otherwise. Multiple choice problems may have more than one correct answer if that is indicated in the problem statement; be sure to select all that apply. Fill-in problems require a short answer such as a number, word, or phrase.
2.
Fill-In: Suppose \(p(x)\in F[x]\) has \(\alpha\) as a root. Then \(M_{\alpha}(x) \fillinmath{\div} p(x)\text{.}\)
3.
True/False: The factors of \(Q(x)=x^{q^n}-x\) in \(F[x]\) with \(|F|=q\) are exactly the monic irreducible polynomials in \(F[x]\) with degree dividing \(n\) and these are the same as the minimal polynomials of the elements of \(E\) the field of roots of \(Q\) over \(F\text{.}\)
Short Response.
Your responses to these questions should be complete solutions with justifications, as per the Grading Specifications.
4.
In this problem you will investigate the structure of a field \(K\) of order \(2^6\) with primitive element \(\beta\text{.}\) It may be useful to recall that the irreducible polynomials of degree 4 or lower over \(\F_2\) are
\begin{gather*}
x, x+1\\
x^2+x+1 \\
x^3+x+1, x^3+x^2+1 \\
x^4+x+1, x^4+x^3+1, x^4+x^3+x^2+x+1 \text{.}
\end{gather*}
(a)
What sizes of subfields can \(K\) have?
(b)
What degrees can minimal polynomials of elements of \(K\) have?
(c)
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 \(K\text{.}\)
Hint.
Solution.
These polynomials are the degree 1 and 2 monic irreducibles:
\begin{equation*}
x^4-x=x(x+1)(x^2+x+1)\text{.}
\end{equation*}
The corresponding elements are \(0,1\) and \(\beta^{21}\) and \(\beta^{42}\text{,}\) since these two elements have order 3 and form a conjugacy class of size 2 (since \(\beta^{21}^4=\beta^{84}=\beta^{21}\) ).
(d)
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 \(K\text{.}\)
Solution.
Similarly, these are the degree 1 and 3 monic irreducibles:
\begin{equation*}
x^8-x=x(x+1)(x^3+x+1)(x^3+x^2+1)\text{.}
\end{equation*}
The corresponding elements are \(0,1\) and the elements of order \(7\) which are the \(\beta^{i}\) such that \(\gcd(i,63)=9\text{.}\)
\begin{equation*}
\{\beta^9, \beta^{18}, \beta^{36}\}, \quad \{\beta^{27},\beta^{54},\beta^{45}\}\text{.}
\end{equation*}
(e)
What is the intersection of these two subfields?
(f)
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 \(\GF(2)\text{?}\)
Solution.
There are \(10\) elements in the union of the two fields above, so the remaining \(54\) elements of \(K\) have minimal polynomials of degree \(g\) (i.e. don’t lie in a proper subfield). Since each such polynomial has \(6\) roots in \(K\) and all monic irreducible polynomials of degree \(6\) are minimal polynomials of some element in \(K\) we see that there must be a total of \(9\) of
5.
Let \(K\) be the field \(\GF(2^3)\) and let \(\beta\) be an element of multiplicative order 9 in the extension field \(E\) of \(K\) with extension degree \([E:K]=2\text{.}\)
(a)
Partition the set of powers
\begin{equation*}
\{1,\beta,\beta^2,\dots,\beta^8\}
\end{equation*}
into conjugacy classes (with respect to \(K\)).
(b)
For each element \(\beta^i\text{,}\) find the degree of the minimal polynomial \(M_{\beta^i}(x)\) (with respect to \(K\)).
(c)
For each minimal polynomial \(M_{\beta^i}(x)\text{,}\) find its constant coefficient (i.e., the value of \(M_{\beta^i}(x)\) at \(x=0\)).
(d)
Solution.
For each \(1\leq i \leq 8\text{,}\) we have
\begin{equation*}
M_{\beta^i}(x)=(x-\beta^i)(x-\beta^{i})=x^2-(\beta^i+\beta^{-i})x+1 \text{.}
\end{equation*}
The coefficients of the minimal polynomial of \(\beta^i\) with respect to \(K\) are in \(K\text{,}\) so \(\beta^i+\beta^{-i}\) is in \(K\text{.}\) For \(i=0\) we have \(1+1=0\in K\text{.}\)
(e)
Find the value of \(\beta^3+\beta^{-3}\text{.}\)
Solution.
Notice that \(\beta^3\) and \(\beta^{-3}\) are elements of order \(3\) in a field of size \(64\text{.}\) Therefore the set \(\{0,1,\beta^3,\beta^{-3}\}\) is the subfield of size \(4\) in \(E\) (and thus also in \(K\)). In this field we must have \(\beta^3+\beta{-3}=1\text{,}\) since any other possibility forces either \(\beta^3=0, \beta^{-3}=0,\) or \(\beta^3=\beta^{-3}\text{,}\) which are all not true.
(f)
Identify the minimal polynomials \(M_{\beta^i}(x)\) (with respect to \(K\)) whose coefficients are in \(\GF(2)\) and explicitly give their coefficients.
Solution.
These polynomials are those where the conjugacy class of powers of \(\beta\) are the same over \(K\) and over \(\GF(2)\text{,}\) i.e. they are the powers of \(\beta\) for which we have
\begin{equation*}
\{\beta^{2i}\mid i \geq 0\} = \{\beta^{8i} \mid i \geq 0\}\text{.}
\end{equation*}
Certainly \(M_1(x)=x-1\) is such a polynomial. The other such is \(M_{\beta^3}(x)=M_{\beta^6}(x)\) since this is the only higher power of \(\beta\) where we have \(\beta^{2i}=\beta^{8i}\) (by inspection from (a)). From the last part (or because it is the minimal polynomial of an element of \(\GF(4)\setminus\GF(2)\) we have \(M_{\beta^3}(x)=x^2+x+1\text{.}\)
