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 credits 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.
Let \(a(x)\) be a monic irreducible polynomial over a finite field \(F\text{.}\) Show that \(a(x)\) is the minimal polynomial of the element \(\xi\) in the extension field \(F[\xi]/a(\xi).\)
Let \(\alpha\) be a nonzero element in an extension field \(E\) of \(F=\GF(q)\text{.}\) Show that the conjugacy classes of \(\alpha\) and \(\alpha^{-1}\) (with respect to \(F\)) are the same size.
Let \(M_{\alpha}(x)=\sum_{i=0}^m a_i x^i\) be the minimal polynomial of \(\alpha\) as in (b) (with respect to \(F\)). Show that the minimal polynomial of \(\alpha^{-1}\) is given by
That is, up to scaling by some nonzero element of \(F\text{,}\) the minimal polynomial of \(\alpha^{-1}\) is the reciprocal polynomial obtained by reversing the order of coefficients of the minimal polynomial of \(\alpha\text{.}\)
Let \(\varphi: K \to E\) be an isomorphism (a multiplicative, additive, bijective map). Show that the minimal polynomial (with respect to \(F\)) of every element \(\beta \in K\) is the same as the minimal polynomial of the element \(\varphi(\beta)\in E\text{.}\)
Show that every isomorphism \(\varphi: K \to E\) is completely defined by the value of \(\varphi\) at an element \(\beta\in K\) that does not belong to any proper subfield of \(K\text{.}\)
Given an element \(\beta \in K\) that does not belong to any proper subfield of \(K\text{,}\) what are the possible values that \(\varphi(\beta)\) may take?
Let \(C\) be a binary \([15,9]\) cyclic code generated by \(g(x)=1+x^3+x^4+x^5+x^6\text{.}\) Suppose that we know \(C\) can correct any burst of length 3 or less. Decode each of the following vectors.
Let \(\alpha\) be a primitive element in \(E=\GF(2^m)\) and let \(C_{\BCH}\) be a narrow-sense primitive BCH code (of length \(n=2^m-1\)) over \(F=\GF(2)\) whose parity-check matrix is obtained by representing each entry in
