Textbook __Abstract Algebra__, by W. E. Deskins.
Dover, 1992. ISBN: 0-486-68888-7 (pbk.) The library copy is on closed reserve
in the Sci/Eng Library.

1. Textbook, p. 255, #7. Denote as usual by Ö
2 the positive real number whose square is 2. Show that the set a + bÖ
2, where a and b are rational, is a field relative to the usual operations.

2. Textbook, p.256, #15. Let **C** = **R **x
**R**, (where **R** is the real numbers) and define operations as
follows:

(a, b) + (c, d) = (a + c, b + d) and (a, b)(c, d) = (ac - bd, ad + bc).

Show that the resulting system is a field. (It is
generally known as the field of complex numbers. The element (0, 1) is
usually denoted by i and the element (a, b) by a + bi.)

3. Let D = {a_{1}, a_{2}, ... a_{n}}
be an integral domain with only finitely many elements. Let b be an arbitrary
but fixed nonzero element in D.

(a) Define a function a
from D into D by a
(a_{i}) = a_{i}b for every a_{i} in D. Show that
the function a is
1 - 1.

(b) Explain why the finiteness of D guarantees that a must be onto simply because it is 1 - 1.

(c) The image of one of the a_{j}'s is the
multiplicative identity 1 of D. Suppose that a
(a_{i}) = 1. Then why is a_{i} a multiplicative inverse
for b?

By (c) the element b has a multiplicative inverse. But b could have been any nonzero element. Therefore, every nonzero element of D has a multiplicative inverse, and D is a field.