Math 361, Spring 2013, Assignment 1

By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so worthy of my consideration.

- Mary Shelley, Frankenstein

Carefully define the following terms, then give one example and one non-example of each:

1. Ring
2. Ring with unity
3. Integral domain
4. Field

Carefully state the following theorems (you need not prove them):

1. Universal mapping property of the field of fractions (as given in class).

Solve the following problems:

1. Let \(D\) be the ring of Gaussian integers, i.e. the set of complex numbers \(D = \{a + bi | a, b\in\mathbb{Z}\}\).

   (a) Show that \(D\) is an integral domain.

   (b) Decide whether \(D\) is a field. Prove your answer.

   (c) Let \(F\) be the field of fractions of \(D\), and let \(\phi:D\rightarrow\mathbb{C}\) be the inclusion map. Because of the universal mapping property of \(F\), the inclusion \(\phi\) induces an embedding \(\psi:F\rightarrow\mathbb{C}\). Describe the image of this embedding. (Hint: this amounts to asking exactly which complex numbers can be written in the form \((a+bi)/(c+di)\) where \(a, b, c, d\) are integers.)

2. Let \(D\) be a field. Show that the field of fractions of \(D\) is isomorphic to \(D\) itself. (Hint: what is the image of the embedding \(\iota\)?)

--------------------End of assignment--------------------