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.
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.
Problem Specs/Notes: This problem needs you to carefully distinguish maps that are ring homomorphisms from those that are only group homomorphisms for a Success.
For any positive integer \(n\text{,}\) how many polynomials are there of degree \(n\) over \(\Z_2\text{?}\) How many distinct polynomial functions from \(\Z_2\) to \(\Z_2\) are there?
Let \(\ev_1\) be the ring homomorphism from \(\Z[x]\) to \(\Z\) given by \(\ev_1(f)=f(1)\text{.}\) Find a polynomial \(g \in \Z[x]\) such that \(\ker \ev_1 =\langle g\rangle\text{.}\) Is there more than possibility for \(g(x)\text{?}\) To what familiar ring is \(\Z[x]/\ker \ev_1\) isomorphic?
Problem Specs/Notes: This problem can reference some of your work in the last problem, but be sure to make clear the differences from moving to \(\Q[x]\) for a Success.
