Examples of nonstandard Euclidean functions on Euclidean domain
$begingroup$
An integral domain $R$ is a Euclidean domain iff there exists a function $N: Rsetminus{0} rightarrow mathbb{Z}$ such that
- If $a,bin R$, then there exists $qin R$ such that either $a=qb$ or $N(a-qb)<N(b)$.
- If $a,b in R$, then $N(a)leq N(ab)$.
I understand that condition 2 is unnecessary, as the existence of a function satisfying condition 1 implies the existence of a function satisfying both conditions.
To understand this better, I'd like to see examples of
- A Euclidean function on some Euclidean domain satisfying condition 1 but not condition 2.
- A Euclidean domain with two Euclidean functions $f$ and $g$ satisfying conditions 1 and 2, such that the orderings $f(x)leq f(y)$ and $g(x) leq g(y)$ are nonisomorphic.
Can someone give nice examples of these?
abstract-algebra euclidean-algorithm euclidean-domain
edited Jan 2 at 22:35
user26857
39.3k124183
asked Oct 20 '15 at 1:33
Joshua MeyersJoshua Meyers
796413
$endgroup$
add a comment |
$begingroup$
An integral domain $R$ is a Euclidean domain iff there exists a function $N: Rsetminus{0} rightarrow mathbb{Z}$ such that
- If $a,bin R$, then there exists $qin R$ such that either $a=qb$ or $N(a-qb)<N(b)$.
- If $a,b in R$, then $N(a)leq N(ab)$.
I understand that condition 2 is unnecessary, as the existence of a function satisfying condition 1 implies the existence of a function satisfying both conditions.
To understand this better, I'd like to see examples of
- A Euclidean function on some Euclidean domain satisfying condition 1 but not condition 2.
- A Euclidean domain with two Euclidean functions $f$ and $g$ satisfying conditions 1 and 2, such that the orderings $f(x)leq f(y)$ and $g(x) leq g(y)$ are nonisomorphic.
Can someone give nice examples of these?
abstract-algebra euclidean-algorithm euclidean-domain
edited Jan 2 at 22:35
user26857
39.3k124183
asked Oct 20 '15 at 1:33
Joshua MeyersJoshua Meyers
796413
$endgroup$
add a comment |
$begingroup$
An integral domain $R$ is a Euclidean domain iff there exists a function $N: Rsetminus{0} rightarrow mathbb{Z}$ such that
- If $a,bin R$, then there exists $qin R$ such that either $a=qb$ or $N(a-qb)<N(b)$.
- If $a,b in R$, then $N(a)leq N(ab)$.
I understand that condition 2 is unnecessary, as the existence of a function satisfying condition 1 implies the existence of a function satisfying both conditions.
To understand this better, I'd like to see examples of
- A Euclidean function on some Euclidean domain satisfying condition 1 but not condition 2.
- A Euclidean domain with two Euclidean functions $f$ and $g$ satisfying conditions 1 and 2, such that the orderings $f(x)leq f(y)$ and $g(x) leq g(y)$ are nonisomorphic.
Can someone give nice examples of these?
abstract-algebra euclidean-algorithm euclidean-domain
edited Jan 2 at 22:35
user26857
39.3k124183
asked Oct 20 '15 at 1:33
Joshua MeyersJoshua Meyers
796413
$endgroup$
An integral domain $R$ is a Euclidean domain iff there exists a function $N: Rsetminus{0} rightarrow mathbb{Z}$ such that
- If $a,bin R$, then there exists $qin R$ such that either $a=qb$ or $N(a-qb)<N(b)$.
- If $a,b in R$, then $N(a)leq N(ab)$.
I understand that condition 2 is unnecessary, as the existence of a function satisfying condition 1 implies the existence of a function satisfying both conditions.
To understand this better, I'd like to see examples of
- A Euclidean function on some Euclidean domain satisfying condition 1 but not condition 2.
- A Euclidean domain with two Euclidean functions $f$ and $g$ satisfying conditions 1 and 2, such that the orderings $f(x)leq f(y)$ and $g(x) leq g(y)$ are nonisomorphic.
Can someone give nice examples of these?
abstract-algebra euclidean-algorithm euclidean-domain
abstract-algebra euclidean-algorithm euclidean-domain
edited Jan 2 at 22:35
user26857
39.3k124183
asked Oct 20 '15 at 1:33
Joshua MeyersJoshua Meyers
796413
edited Jan 2 at 22:35
user26857
39.3k124183
asked Oct 20 '15 at 1:33
Joshua MeyersJoshua Meyers
796413
edited Jan 2 at 22:35
user26857
39.3k124183
edited Jan 2 at 22:35
user26857
39.3k124183
edited Jan 2 at 22:35
user26857
39.3k124183
39.3k124183
asked Oct 20 '15 at 1:33
Joshua MeyersJoshua Meyers
796413
asked Oct 20 '15 at 1:33
Joshua MeyersJoshua Meyers
796413
asked Oct 20 '15 at 1:33
Joshua MeyersJoshua Meyers
796413
796413
add a comment |
add a comment |
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