189-235A: Basic Algebra I Midterm Exam

Friday, October 22

1. Let (u_{n})_{n≥0} be the sequence of real numbers defined recursively by the rule

u_{0} = 0, u_{n+1} = 2u_{n} + 1.

Show that u_{n} = 2^{n} − 1 for all n ≥ 0.

2. Compute the greatest common divisor of 121 and 77 and express the result as a linear combination of 121 and 77.

3. Solve the congruence equation 6x ≡ 10 (mod 14).

4. Show that if p ∈ Z is a prime, then the ring Z_{p} of congruence classes modulo p is a field.

5. Give an example of two finite rings R_{1} and R_{2} which have the same cardinality but are not isomorphic. (You should justify your assertion.)

6. Show that the ring C of complex numbers is not isomorphic to the Cartesian product R × R of the real numbers with itself. The next two problems are Bonus Questions

7. Let f be a polynomial in Z[x] of degree d and let p ∈ Z be a prime number. Show that the set

S = {n ∈ Z such that p divides f(n)}

is the union of at most d congruence classes modulo p.

8. Let p = 2m + 1 be an odd prime. Show that 1^{1} · 2^{2} · 3^{3} ⋯ (p − 1)^{p−1} ≡ (−1)^{[m/2]}m! (mod p).