Mathematics 582   Problem Set 1   Due Wednesday, October 2

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 = {a1, a2, ... an} 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 (ai) = aib for every ai 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 aj's is the multiplicative identity 1 of D. Suppose that a (ai) = 1. Then why is ai 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.