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Closed book. You are permitted a calculator and one 8½" ´11" sheet of handwritten notes (both sides). Be sure to show your work.

Find the least-squares fit of ax + b = y for the data

x y
2 7
4 11
5 17
7 23
10 30
10 31
What is the check digit for the ISBN that starts 0-423-14237-?
Decode 110110010010101 which was encoded using the generator g(x) = x⁴ + x + 1 in Z₂[x]/(x¹⁵ + 1).
List all the elements of Z₂[x]/(x³ + x + 1).
Let g(x) be the generator of a linear cyclic code of length. Prove that g(x) | xⁿ + 1.
Let p(x) be a polynomial with rational coefficients. Prove that p(a) = 0 if and only if (x - a) | p(x) (note that "if and only if" means you must prove both that p(a) = 0 implies (x - a) | p(x) and also that (x - a) | p(x) implies p(a) = 0).
Short answer questions:
- Is Z₂[x]/(x³ + 1) a field? Explain why or why not.
- Give an example of a non-commutative ring. What standard technique from secondary school algebra fails in a non-commutative ring?
Given any domain D, we constructed a fraction field F for the domain and proved that D Ì F in class. Explain this process. You don’t have to reproduce all the details of the proofs, but you do have to list all the important steps in both the construction and the proof.