Linear Algebra/Topic: Fields/Solutions

Solutions Problem 1 Show that the real numbers form a field. Answer These checks are all routine; most consist only of remarking that property is so familiar that it does not need to be proved. Problem 2 Prove that these are fields. <li> The rational numbers <li> The complex numbers Answer For both of these structures, these checks are all routine. As with the prior question, most of the checks consist only of remarking that property is so familiar that it does not need to be proved. Problem 3 Give an example that shows that the integer number system is not a field. Answer There is no multiplicative inverse for so the integers do not satisfy condition 5. Problem 4exer:BinField Consider the set subject to the operations given above. Show that it is a field. Answer These checks can be done by listing all of the possibilities. For instance, to verify the commutativity of addition, that , we can easily check it for all possible pairs , , because there are only four such pairs. Similarly, for associativity, there are only eight triples , , , and so the check is not too long. (There are other ways to do the checks, in particular, a reader may recognize these operations as arithmetic "mod ".) Problem 5 Give suitable operations to make the set a field. Answer These will do. As in the prior item, the check that they satisfy the conditions can be done by listing all of the cases, although this way of checking is somewhat long (making use of commutativity is helpful in shortening the work).