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.)

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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

The field of fractions consists of all complex numbers of the form $\frac{a+bi}{c+di}$ with $a,b,c,d\in\mathbb{Z}$ and $c,d$ not both zero. You can show that this is the same as the answer in the back of the book by giving a mutual containment argument. So, first show that $\frac{a+bi}{c+di}$ can be rewritten in the form $q_1+q_2i$ with $q_1,q_2\in\mathbb{Q}$. Then show that $q_1+q_2i$ can be rewritten in the form $\frac{a+bi}{c+di}$ with $a,b,c,d\in\mathbb{Z}$. Steven.Jackson (talk) 20:00, 4 February 2017 (EST)