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

Book:Abstract Algebra Fourth Edition John A. Beachy

Chapter:4 to 6.21 attachmentsSlide 1 of 1

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UNFORMATTED ATTACHMENT PREVIEW

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,

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