Math 361, Spring 2017, Assignment 2

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

1. Formal fraction.
2. Equivalence (of formal fractions).
3. Fraction.
4. $\mathrm{Frac}(D)$.
5. Polynomial function.
6. Polynomial expression.

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

1. Universal mapping property of $\mathrm{Frac}(D)$.

Solve the following problems:

1. Section 21, problems 1 and 2. (In both problems, you are being asked to use the universal mapping property to find a "concrete model" of the field of fractions, as we did in class.)
2. Section 22, problems 1 and 3.
3. Prove Euler's theorem. (Hint: Since $\mathrm{gcd}(a,n)=1$, we can regard $a$ as an element of the group of units $G(\mathbb{Z}_n)$. The order of this group is $\phi(n)$. Now see Theorem 10.12 on page 101 of the text.)

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

Questions:

I am having trouble with describing the field F of quotients of the integral subdomain $D=\{n+mi|n,m \in \mathbb{Z}\}$ of complex numbers. Describe means give the elements of the complex numbers that make up the field of quotients of D in the complex numbers. (The elements of D are the Gaussian integers.) The answer in the back of the book says $\{q_1 + q_2i|q_1, q_2 \in \mathbb{Q}\}$ I want to understand how to get to the answer what are the key steps