Chapter 46

The function \(v\) for \(\mathrm{Z}[x]\) given by \(v(f(x))=(\) degree of \(f(x))\) for \(f(x) \in \mathrm{Z}[x], f(x) \neq 0\)

The function \(v\) for \(\mathrm{Z}[x]\) given by \(v(f(x))=(\) the absolute value of the coefficient of the highes degree nonzero term of \(f(x))\) for nonzeto \(f(x) \in \mathbb{Z}(x]\) \\}

Find a gcd of 49,349 and 15,555 in \(Z\).

Describe how the Euclidean Algorithm can be used to find the ged of \(n\) members \(a_{1}, a_{2}, \cdots, a_{n}\) of a Euclidean domain.

Let us consider \(\mathbf{Z}[x]\) a. Is \(\mathbb{Z}[x]\) a UFD? Why? b. Show that \(\\{a+x f(x)|a \in 2 Z, f(x) \in Z[x]|\) is an ideal in \(Z[x]\). c. Is \(Z[x]\) a PID? (Consider part (b).) d. Is \(\mathbb{Z}(x]\) a Euclidean domain? Why?

Let us consider \(\mathbf{Z}[x]\). a. Is \(\mathrm{Z}[x]\) a UFD? Why? b. Show that \(\\{a+x f(x) \mid a \in 2 Z, f(x) \in \mathbb{Z}[x]\\}\) is an ideal in \(Z[x]\). c. Is \(\mathbf{Z}[x]\) a PID? (Consider part (b).), d. Is \(\mathbf{Z}[x]\) a Eaclidean domain? Why?

Mark each of the following true or false. a. Every Euclidean domain is a PID. b. Every PID is a Euclidean domain. c. Every Euclidean domain is a UFD. d. Every UFD is a Euclidean domain. e. A ged of 2 and 3 in \(Q\) is \(\frac{1}{2}\). f. The Euclidean algorithm gives a constructive method for finding a ged of two integers. g. If \(v\) is a Euclidean norm on a Euclidean domain \(D\), then \(v(1) \leq v\\{a)\) for all nonzero \(a \in D\).h. If \(v\) is a Euclidean norm on a Euclidean domain \(D\), then \(v(1)

Does the choice of a particular Euclidean norm v on a Euclidean domain \(D\) influence the arithmetic structure of \(D\) in any way? Explain.

Let \(D\) be a Euclidean domain and let \(v\) be a Euclidean norm on \(D\). Show that if \(a\) and \(b\) are associates in \(D\). then \(v(a)=v(b)\).

Prove or disprove the following statement: If \(\nu\) is a Euclidean norm on Euclidean domain \(D\), then \(\mid a \in\) \(D \mid v(a)>v(1)] \cup\\{0\\}\) is an ideal of \(D .\)

Show that every field is a Eaclidean domain.

Let \(v\) be a Euclidean norm on a Euclidean domain \(D\). a. Show that if \(s \in \mathbb{Z}\) such that \(s+v(1)>0\), then \(\eta: D^{*} \rightarrow Z\) defined by \(n(a)=v(a)+s\) for nonzero \(a \in D\) is a Euclidean norm on \(D\). As usual, \(D^{*}\) is the set of nonzero elements of \(D\). b. Show that for \(t \in \mathbf{Z}^{+}, \lambda: D^{*} \rightarrow \mathbf{Z}\) given by \(\lambda(a)=t-v(a)\) for nonzero \(a \in D\) is a Euclidean norm on \(D\). c. Show that there exists a Euclidean norm \(\mu\) on \(D\) such that \(\mu(1)=1\) and \(\mu(a)>100\) for all nonzero nonunits \(a \in D .\)

Let \(D\) be a UFD. An element \(c\) in \(D\) is a least common multiple (abbreviated lem) of two elements \(a\) and \(b\) in \(D\) if \(a|c, b| c\) and if \(c\) divides every element of \(D\) that is divisible by both \(a\) and \(b\). Show that every two nonzero elements \(a\) and \(b\) of a Euclidean domain \(D\) have an lem in \(D\). [Htnt: Show that all common multiples, in the obvious sense, of both \(a\) and \(b\) form an ideal of \(D .]\)