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##### Ring Fields

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**Question**

Question;A. Use the;definition for a ring to prove that Z7 is a ring under the operations + and ?;defined as follows;[a]7 + 7 = [a + b]7 and [a]7 ? 7 = [a ? b]7;Note: On the right-hand-side of these equations, + and ? are the usual;operations on the integers, so the modular versions of addition and;multiplication inherit many properties from integer addition and;multiplication.;1. State each step of your proof.;2. Provide written justification for each step of your proof.;B. Use the definition for an integral domain to prove that Z7 is an integral;domain.;1. State each step of your proof.;2. Provide written justification for each step of your proof.;C. Let G be the set of the fifth roots of unity.;1. Use de Moivre?s formula to verify that the fifth roots of unity form a group;under complex multiplication, showing all work.;2. Prove that G is isomorphic to Z5 under addition by doing the following;a. State each step of the proof.;b. Justify each of your steps of the proof.;D. Let F be a field. Let S and T be

Paper#60427 | Written in 18-Jul-2015

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