Abstract Algebra Problems

Time : 8:30 pm to 11:30 pm Feb 23 AST

Book:Abstract Algebra Fourth Edition John A. Beachy

Chapter:4 to 6.2

Instructions
1. You may use a calculator to help you with simple arithmetic (+,-, X,/). No other compu-
tational aids are permitted.
2. You must work entirely on your own. You may consult the course textbook (Beachy/Blair)
and your own notes, but no other sources (online or otherwise) are permitted.
3. Show all your work.
4. You will have 3 hours in which to complete the test and scan and submit your answers.
Problems
1. Factor f(x) = 25 – 24 – 6×2 +8x – 2 completely into irreducibles in Q[x] and Z3[r]. Indicate
how you know your factors are irreducible.
2. Short answer (no justification required):
(a) Identify all units and all zero-divisors in the ring Z x Z12.
(b) Give an example of a nonzero ideal in Z[x] that is prime but not maximal.
(c) Give an example of a field with exactly 25 elements.
3. For an ideal I of a ring R, define VI := {a E R : q” E I for some n >0}.
(a) Prove that VI is an ideal of R.
(b) Find 72Z in the ring Z.
4. Throughout this problem, let R=Z[V2] and let I = (2) R.
(a) List all distinct elements of R/I.
(b) Is I a prime ideal? Explain.
(c) Use the First Isomorphism Theorem to prove that R/I is isomorphic to the ring
s={C9): ,vcz}
Be sure to establish that your map from R to S is indeed a ring homomorphism.
(d) Prove that R/I is not isomorphic to any of Z4, Z2 x Z2, or Z2[2]/(x2 + x + 1). This
requires only a very brief rationale for each.