1.
True/False: If \(F\) is a finite field, there are polynomials in \(F[x]\) of every degree \(s\geq 1\) that are irreducible over \(F\text{.}\)
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Worksheet Weekly Practice 6
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: A polynomial \(p(x)\in F[x]\) is irreducible over \(F\) if \(p(x)=a(x)b(x)\) implies that .
3.
Fill-In: (Bezout’s Identity). Let \(f(x),g(x)\in F[x]\text{,}\) not both zero. Then there exist \(s(x),t(x)\in F[x]\) such that
\begin{equation*}
\fillinmath{s(x)f(x)+t(x)g(x)=\gcd(f(x),g(x))}\text{.}
\end{equation*}
Short Response.
Your responses to these questions should be complete solutions with justifications, as per the Grading Specifications.
4.
Find all irreducible quadratic and cubic polynomials over \(F=GF(3)\text{.}\) Clearly explain why the polynomials you list are irreducible and why the polynomials not on your list are reducible.
Solution.
The irreducible quadratics are those without a factor of \(x,x+1\text{,}\) or \(x+2\text{.}\) The monic reducible ones are therefore
\begin{align*}
x^2 \amp \amp (x+1)^2\amp=x^2+2x+1 \amp (x+2)^2\amp=x^2+x+1 \\
x(x+1)\amp =x^2+x \amp x(x+2)\amp=x^2+2x \amp (x+1)(x+2)\amp=x^2+2
\end{align*}
The other three monic quadratics are thus irreducible, as are those obtained by multiplying these by \(2\text{:}\)
\begin{equation*}
x^2+2x+2, \qquad x^2+x+2, \qquad x^2+1
\end{equation*}
Similarly, the irreducible cubics are those without a factor that is either linear or one of the three irreducible quadratics. We can get cubics with those factors by either cubing one of the three linear factors or multiplying one linear and one quadratic. So there are a total of 12 reducible monic cubics and thus the other 15 are irreducible along with the 15 obtained by multiplying them by \(2\text{.}\)
5.
Let \(f(x)=x^3+x+1\) and \(g(x)=x^5+x^4+2x^2+x+2\) in \(GF(3)[x]\text{.}\) Compute \(d(x)=\gcd(f(x),g(x))\) and polynomials \(s(x),t(x)\) satisfying Bezout’s Identity for \(f(x)\) and \(g(x)\text{.}\)
Solution.
We have
\begin{align*}
x^5+x^4+2x^2+x+2 \amp = (x^2+x+2)(x^3+x+1) + x \\
x^3+x+1 \amp = (x^2+1)x+ 1 \text{.}
\end{align*}
So \(d(x)=\gcd(f(x),g(x))=1.\) Reversing the steps gives
\begin{equation*}
(x^4+x^3+x)(x^3+x+1)+(2x^2+2)(x^5+x^4+2x^2+x+2)=1\text{.}
\end{equation*}
So \(s(x)=x^4+x^3+x\) and \(t(x)=2x^2+2\text{.}\)
