Math 361, Spring 2021, Assignment 8

Read:

1. Section 21.
2. Section 22.

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

1. Addition (of fractions).
2. Multiplication (of fractions).
3. $\mathrm{Frac}(D)$ (the field of fractions of the integral domain $D$).
4. Canonical injection (of an integral domain $D$ into its field of fractions).
5. Polynomial function (from a ring $R$ into itself).
6. Polynomial expression (with coefficients in a ring $R$).
7. Addition (of polynomial expressions).
8. Multiplication (of polynomial expressions).
9. $R[x]$ (the ring of polynomial expressions, with coefficients in $R$, in the indeterminate $x$, or "$R$ adjoin $x$" for short).
10. $R[x,y]$ (the ring of bivariate polynomials in the indeterminates $x$ and $y$).
11. $R[x_1,\dots,x_n]$ (the ring of multivariate polynomials in the indeterminates $x_1,\dots,x_n$).

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. (Concrete models of $\mathrm{Frac}(D)$). Let $D$ be any integral domain, and suppose that $\phi:D\rightarrow F$ is any injection of $D$ into a field. Let $\widehat{\phi}:\mathrm{Frac}(D)\rightarrow F$ be the injection described by the universal mapping property. Show that $\mathrm{im}\left(\widehat{\phi}\right)$ is an isomorphic "copy" of $\mathrm{Frac}(D)$ inside $F$. (Hint: this problem is much easier than it looks). In subsequent problems we refer to this copy as the concrete model of $\mathrm{Frac}(D)$ arising from the injection $\phi$.
2. (Concrete model of $\mathbb{Q}$ inside $\mathbb{R}$). Recall that we have officially defined the symbol $\mathbb{Q}$ to mean $\mathrm{Frac}(\mathbb{Z})$. Let $\phi:\mathbb{Z}\rightarrow\mathbb{R}$ denote the "initial morphism" defined in Assignment 5. Since $\mathrm{char}(\mathbb{R})=0$, this is an injection of $\mathbb{Z}$ into $\mathbb{R}$. Describe the concrete model of $\mathbb{Q}$ arising from this injection. (Hint: this problem is much harder than it looks. Your high school teacher has perhaps told you the answer: it is the set of real numbers having decimal expansions that are "eventually periodic." Proving this is surprisingly tricky; you will need both the elementary-school integer division algorithm and a clever application of the theory of the geometric series that you learned in Calculus II.)
3. Section 21, problems 1 and 2 (translation of problems: you are being asked to describe the concrete model of the field of fractions of the given integral domain $D$ arising from the inclusion of $D$ in the given field $F$).
4. Suppose that the sequence $a=(a_0,a_1,\dots)$ has only finitely many non-zero entries, say $a_i=0$ for all $i>N$, and similarly that the sequence $b=(b_0,b_1,\dots)$ satisfies $b_j=0$ for all $j>M$. Let $(c_0,c_1,\dots)$ be the product $ab$ that we defined in class, i.e. $c_k=\sum_{i+j=k}a_ib_j$. Show that $c_k=0$ for all $k>N+M$, so that multiplication of polynomial expressions is really a well-defined binary operation. (Hint: show that if $k>N+M$, then in every term of the formula defining $c_k$ we must have either $i>N$ or $j>M$. It is easiest to prove this in contrapositive form.)
5. Section 22, problems 1, 2, 3, 4, 5, and 6.
6. (Rational expressions). Next week we shall prove that whenever $D$ is an integral domain, so is $D[x]$. For purposes of this exercise, you may take this fact for granted. Thus, the field of fractions of $D[x]$ is a well-defined object, which is usually denoted $D(x)$. Write down two "random" elements of the field $\mathbb{R}(x)$, and show how to add them, and also how to multiply them.
7. (Multivariate rational expressions) In the last exercise we asserted that whenever $D$ is an integral domain, so is $D[x]$. Iterating this principle, we see that $D[x,y]$ is also an integral domain, as is $D[x,y,z]$, and indeed (by induction) $D[x_1,\dots,x_n]$. Thus, the field of fractions $\mathrm{Frac}(D[x_1,\dots,x_n])$ is a well-defined object, usually denoted $D(x_1,\dots,x_n)$. Write down two "random" elements of the field $\mathbb{R}(x,y)$, and show how to add them, and also how to multiply them.

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