Math 361, Spring 2016, Assignment 2

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

1. (Unital) embedding (of one unital ring into another).
2. Field of fractions (of an integral domain).
3. Polynomial (with coefficients in a commutative unital ring $R$).
4. Addition (of polynomials).
5. Multiplication (of polynomials).

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

1. Theorem characterizing which rings can be embedded in fields.
2. Uniqueness theorem for fields of fractions.
3. Existence theorem for fields of fractions.

Solve the following problems:

In the following problems, let $D$ be the set of complex numbers whose real and imaginary parts are both integers, i.e. $D=\{n+mi\,|\,n,m\in\mathbb{Z}\}$. (The elements of $D$ are called Gaussian integers.)

1. Prove that $D$ is an integral domain. (Hint: it is a unital subring of $\mathbb{C}$.)
2. Give two or three concrete sample calculations in the field of fractions $F(D)$.
3. Let $\phi:D\rightarrow\mathbb{C}$ denote the inclusion map. Then $\phi$ is an embedding of $D$ into a field, so we have a unique embedding $\widehat{\phi}:F(D)\rightarrow\mathbb{C}$ making $\widehat{\phi}\circ\iota = \phi$. Describe the image of $\widehat{\phi}$. (This is a subfield of $\mathbb{C}$ isomorphic to $F(D)$.)
4. Section 22, problems 1, 3, and 5.

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