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.

1. The rational numbers $\mathbb {Q}$
2. The complex numbers $\mathbb {C}$

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 $2$ so the integers do not satisfy condition 5.

Problem 4

Consider the set $\{0,1\}$ 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 $a+b=b+a$, we can easily check it for all possible pairs $a$, $b$, because there are only four such pairs. Similarly, for associativity, there are only eight triples $a$, $b$, $c$, 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 $2$".)

Problem 5

Give suitable operations to make the set $\{0,1,2\}$ a field.

Answer

These will do.

${\begin{array}{c|ccc}+&0&1&2\\\hline 0&0&1&2\\1&1&2&0\\2&2&0&1\end{array}}\qquad {\begin{array}{c|ccc}\cdot &0&1&2\\\hline 0&0&0&0\\1&0&1&2\\2&0&2&1\end{array}}$

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).